aa r X i v : . [ m a t h . OA ] O c t DEFINABLE K -HOMOLOGY OF SEPARABLE C*-ALGEBRAS MARTINO LUPINI
Abstract.
In this paper we show that the K-homology groups of a separable C*-algebra can be enriched withadditional descriptive set-theoretic information, and regarded as definable groups . Using a definable version of theUniversal Coefficient Theorem, we prove that the corresponding definable
K-homology is a finer invariant than thepurely algebraic one, even when restricted to the class of UHF C*-algebras, or to the class of unital commutativeC*-algebras whose spectrum is a 1-dimensional connected subspace of R . Introduction
Given a compact metrizable space X , the group Ext ( X ) classifying extensions of the C*-algebra C ( X ) by theC*-algebra K ( H ) of compact operators was initially considered by Brown, Douglas, and Fillmore in their celebratedwork [BDF77]. There, they showed that Ext ( − ) is indeed a group, and that defining, for a compact metrizablespace X , ˜K p ( X ) := (cid:26) Ext ( X ) if p is odd,Ext (Σ X ) if p is even;where Σ X is the suspension of X , yields a (reduced) homology theory that satisfies all the Eilenberg–Steenrod–Milnor axioms for Steenrod homology, apart from the Dimension Axiom; see also [KS77]. They furthermoreobserved, building on a previous insight of Atiyah [Ati70], that such a homology theory can be seen as the Spanier–Whitehead dual of topological K-theory [Ati89].More generally, for an arbitrary separable unital C*-algebra A , one can consider a semigroup Ext ( A ) classifyingthe essential, unital extensions of A by K ( H ). By Voiculescu’s non-commutative Weyl-von Neumann Theorem[Voi76, Arv77], the trivial element of Ext ( A ) correspond to the class of trivial essential, unital extensions. Thegroup Ext ( A ) − of invertible elements of Ext ( A ) corresponds to the essential, unital extensions that are semi-split .Thus, by the Choi–Effros lifting theorem [CE76], Ext ( A ) is a group when A is nuclear. One can extend K-homologyto the category of all separable C*-algebras by settingK p ( A ) = (cid:26) Ext ( A + ) − if p is odd,Ext(( SA ) + ) − if p is even;where SX is the suspension of A and A + is the unitization of A . This gives a cohomology theory on the category ofseparable C*-algebras, which can be recognized as the dual of K-theory via Paschke duality [Pas81, Hig95, KS17].Kasparov’s bivariant functor KK ( − , − ) simultaneously generalizes K-homology and K-theory, where K ( A ) isrecovered as KK ( A, C ) and K ( A ) as KK ( A, C ( R )).It was already noticed in the seminal work of Brown, Douglas, and Fillmore [BDF77, BDF73] that the invariantK p ( X ) for a compact metrizable space X can be endowed with more structure than the purely algebraic groupstructure. Indeed, one can write X as the inverse limit of a tower ( X n ) n ∈ ω of compact polyhedra, and endowK p ( X ) with the topology induced by the maps K p ( X ) → K p ( X n ) for n ∈ ω , where K p ( X n ) is a countable groupendowed with the discrete topology. This gives to K p ( X ) the structure of a topological group, which is however ingeneral not Hausdorff.The study of K p ( A ) as a topological group for a separable unital C*-algebra A was later systematically undertakenby Dadarlat [Dad00, Dad05] and Schochet [Sch01, Sch02, Sch05] building on previous work of Salinas [Sal92]. (Infact, they consider more generally Kasparov’s KK-groups.) In [Sch01, Dad05] several natural topologies on K p ( A ), Date : October 23, 2020.2020
Mathematics Subject Classification.
Primary 19K33, 54H05; Secondary 46M20, 46L80.
Key words and phrases.
K-homology, KK-theory, Universal Coefficient Theorem, C*-algebra, definable group.The author was partially supported by the Marsden Fund Fast-Start Grant VUW1816 from the Royal Society Te Ap¯arangi. corresponding to different ways to define K-homology for separable C*-algebras, are shown to coincide and to turnK p ( A ) into a pseudo-Polish group. This means that, if K p ∞ ( A ) denotes the closure of zero in K p ( A ), then thequotient of K p ( A ) by K p ∞ ( A ) is a Polish group. In [Sch02], for a C*-algebra A satisfying the Universal CoefficientTheorem (UCT), the topology on K p ( A ) is related to the UCT exact sequence, and K p ∞ ( A ) is shown to be isomorphicto the group PExt (K − p ( A ) , Z ) classifying pure extensions of K − p ( A ) by Z . A characterization of K p ∞ ( A ) for anarbitrary separable nuclear C*-algebra A is obtained in [Dad00]. For a separable quasidiagonal C*-algebra satisfyingthe UCT, K ∞ ( A ) is shown to be the subgroup of K ( A ) = Ext ( A + ) corresponding to quasidiagonal extensions of A + by K ( H ) [Sch02]; see also [Bro84] for the commutative case. The quotient K p w ( A ) of K p ( A ) by K p ∞ ( A ) is thegroup KL p ( A, C ) introduced by Rørdam [Rør95]. A universal multicoefficient theorem describing K p w ( A ) in termsof the K-groups of A with arbitrary cyclic groups as coefficients is obtained in [DE02] for all separable nuclearC*-algebras satisfying the UCT; see [Dad05, Theorem 5.4].In many cases of interest, the topology on K p ( A ) turns out to be trivial, i.e. the closure of zero in K p ( A ) isthe whole group. For example, the topology on K ( A ) is trivial when A is a UHF C*-algebra, despite the factthat K ( A ) is not trivial, and in fact uncountable. Similarly, for every 1-dimensional solenoid X , the topology on˜K ( X ) is trivial, although ˜K ( X ) is an uncountable group.In this paper, we take a different approach and consider the group K p ( A ), rather than as a pseudo-Polishtopological group, as a definable group . This should be thought of as a group G explicitly defined as the quotient ofa Polish space X by a “well-behaved” equivalence relation E , in such a way that the multiplication and inversionoperations in G are induced by a Borel functions on X . This is formally defined in Section 1.5, where the notion ofwell-behaved equivalence relation is made precise. The definition is devised to ensure that the category of definablegroups has good properties, and behaves similarly to the category of standard Borel groups. A morphism in thiscategory is a definable group homomorphism, namely a group homomorphism that lifts to a Borel function betweenthe corresponding Polish spaces.It has recently become apparent that several homological invariants in algebra and topology can be seen asfunctors to the category of definable groups. The homological invariants Ext and lim are considered in [BLP20],whereas Steenrod homology and ˇCech cohomology are considered in [BLP, Lup20]. It is shown there that thedefinable versions of these invariants are finer than the purely algebraic versions.In this paper, we show that, for an arbitrary separable C*-algebra A , K p ( A ) can be regarded as a definablegroup. Furthermore, different descriptions of K p ( A )—in terms of extensions, Paschke duality, Fredholm modules,and quasi-homomorphisms—yield naturally definably isomorphic definable groups. For C*-algebras that have aKK-filtration in the sense of Schochet [Sch96], we show that the definable subgroup K p ∞ ( A ) of K p ( A ) is definably isomorphic to PExt (K − p ( A ) , Z ). The latter is regarded as a definable group as in [BLP20, Section 7]. (In fact,PExt (K − p ( A ) , Z ) is the quotient of a Polish group by a Borel Polishable subgroup, and hence a group with aPolish cover in the parlance of [BLP20, Section 7].)Using this and the rigidity theorem for PExt (Λ , Z ) from [BLP20, Section 7] where Λ is a torsion-free abeliangroup without finitely-generated direct summands, we prove that definable K-homology provides a finer invariantthan the purely algebraic (or topological) groups K p ( A ) for a separable C*-algebra A , even when one restricts toUHF C*-algebras or commutative unital C*-algebras whose spectrum is a 1-dimensional subspace of R . Theorem A.
The definable K -group is a complete invariant for UHF C*-algebras up to stable isomorphism. Incontrast, there exists an uncountable family of pairwise non stably isomorphic UHF C*-algebras with algebraicallyisomorphic K -groups (and trivial K -groups). Theorem B.
The definable ˜K -group is a complete invariant for -dimensional solenoids up to homeomorphism. Incontrast, there exists an uncountable family of pairwise non homeomorphic -dimensional solenoids with algebraicallyisomorphic ˜K -groups (and trivial ˜K -groups). The historic evolution in the treatment of K-homology described above should be compared with the similarevolution in the study of unitary duals of second countable, locally compact groups or, more generally, separableC*-algebras. Given a separable C*-algebra A , its unitary dual ˆ A is the quotient of the Polish space Irr( A ) of unitaryirreducible representations of A by the relation of unitary equivalence. This includes as a particular instance thecase of second countable, locally compact groups, by considering the corresponding universal C*-algebras. Whileinitially ˆ A was considered as a topological space endowed with the quotient topology, it was recognized in the seminal EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 3 work of Mackey, Glimm, and Effros [Mac57, Gli61, Eff65] that a more fruitfuil theory is obtained by consideringˆ A endowed with the quotient Borel structure, called the Mackey Borel structure. This led to the notion of type IC*-algebra, which precisely captures those separable C*-algebras with the property that the Mackey Borel structureis standard. It was soon realized that, in the non type I case, the right notion of “isomorphism” of Macky Borelstructures on duals ˆ A, ˆ B corresponds to a bijection ˆ A → ˆ B that is induced by a Borel function Irr ( A ) → Irr ( B ).In our terminology from Section 1.4, this corresponds to regarding a unitary dual ˆ A as a definable set , where anisomorphism of Mackey Borel structures on ˆ A, ˆ B is a definable bijection ˆ A → ˆ B . For example, this approach istaken by Elliott in [Ell77], where he proved that the unitary duals of any two separable AF C*-algebras that arenot type I are isomorphic in the category of definable sets. It is a question of Dixmier from 1967 whether theunitary duals of any two non-type I separable C*-algebras are isomorphic in the category of definable sets; see[Tho15, Far12, KLP10]. This problem was recently considered in the case of groups by Thomas, who showed thatthe unitary duals of any two countable amenable non-type I groups are isomorphic in the category of definable sets[Tho15, Theorem 1.10]. Furthermore, the unitary dual of any countable group admits a definable injection to theunitary dual of the free group on two generators [Tho15, Theorem 1.9].The work of Mackey, Glimm, and Effors on unitary representations pioneered the application of methods fromdescriptive set theory to C*-algebras. More recent applications have been obtained by Kechris [Kec98] and Farah–Toms–T¨ornquist [FTT14, FTT12], who studied the problem of classifying several classes of C*-algebras from theperspective of Borel complexity theory; see also [Lup14, GL16, EFP + Polish spaces and definable groups
In this section we recall some fundamental notions concerning Polish spaces and Polish groups, as well as standardBorel spaces and standard Borel groups, as can be found in [BK96, Kec95, Gao09]. We also consider the notion ofPolish category, which is a category whose hom-sets are Polish spaces and composition of morphisms is a continuousfunction, and establish some of its basic properties. Furthermore, we recall the notion of idealistic equivalencerelation on a standard Borel space and some of its fundamental properties as established in [KM16, MR12]. Wethen define precisely the notion of (semi)definable set and (semi)definable group.1.1.
Polish spaces and standard Borel spaces. A Polish space is a second countable topological space whosetopology is induced by a complete metric. A subset of a Polish space X is G δ if and only if it is a Polish spacewhen endowed with the subspace topology. If X is a Polish space, then the Borel σ -algebra of X is the σ -algebra MARTINO LUPINI generated by the collection of open sets. By definition, a subset of X is Borel if it belongs to the Borel σ -algebra. If X, Y are Polish spaces, then the product X × Y is a Polish space when endowed with the product topology. Moregenerally, if ( X n ) n ∈ ω is a sequence of Polish spaces, then the product Q n ∈ ω X n is a Polish space when endowed withthe product topology. The class of Polish spaces includes all locally compact second countable Hausdorff spaces.We denote by ω the set of natural numbers including 0. We regard ω and any other countable set as a Polish spaceendowed with the discrete topology. The Baire space ω ω is the Polish space obtained as the infinite product ofcopies of ω .A standard Borel space is a set X endowed with a σ -algebra (the Borel σ -algebra) that comprises the Borel setswith respect to some Polish topology on X . A function between standard Borel spaces is Borel if it is measurablewith respect to the Borel σ -algebras. A subset of a standard Borel space X is analytic if it is the image of a Borelfunction f : Z → X for some standard Borel space Z . This is equivalent to the assertion that there exists a Borelsubset B ⊆ X × ω ω such that B = proj X ( A ) is the projection of A on the first coordinate. A subset of X is co-analytic if its complement is analytic. One has that a subset of X is Borel if and only if it is both analytic andco-analytic.Given standard Borel spaces X, Y , we let X × Y be their product endowed with the product Borel structure,which is also a standard Borel space. If ( X n ) n ∈ ω is a sequence of standard Borel spaces, then their disjoint union X is a standard Borel space, where a subset A of X is Borel if and only if A ∩ X n is Borel for every n ∈ ω . Theproduct Q n ∈ ω X n is also a standard Borel space when endowed with the product Borel structure. In the followingproposition, we collect some well-known properties of the category of standard Borel spaces and Borel functions. Proposition 1.1.
Let SB be the category that has standard Borel spaces as objects and Borel functions andmorphisms.(1) If X is a standard Borel space and A ⊆ X is a Borel subset, then A is a standard Borel space when endowedwith the induced standard Borel structure;(2) If X, Y are standard Borel spaces, f : X → Y is an injective Borel function, and A ⊆ X is Borel, then f ( A ) is a Borel subset of Y ;(3) If X, Y are standard Borel spaces, and f : X → Y is a bijective Borel function, then the inverse function f − : Y → X is Borel;(4) If X, Y are standard Borel spaces, and there exist injective Borel functions f : X → Y and g : Y → X , thenthere exists a Borel bijection h : X → Y ;(5) The category SB has finite products, finite coproducts, equalizers, and pullbacks;(6) A Borel function is monic in SB if and only if it is injective, and epic in SB if and only if it is surjective;(7) An inductive sequence of standard Borel spaces and injective Borel functions has a colimit in SB . A Polish group is a topological group whose topology is Polish. If G is a Polish group, and H is a closed subgroupof G , then H is a Polish group when endowed with the subspace topology. If furthermore H is normal, then G/H isa Polish group when endowed with the quotient topology. If G , G are Polish groups, and ϕ : G → G is a Borelfunction, then ϕ is continuous. In particular, if G is a Polish space, then it has a unique Polish group topology thatinduces its Borel structure. A subgroup H of a Polish group G is Polishable if it is Borel and there is a (necessarilyunique) Polish group topology on H that induces the Borel structure on H inherited from G . This is equivalent tothe assertion that H is equal to the range of a continuous group homomorphisms ˆ G → G for some Polish group ˆ G .If G is a Polish group, then a Polish G -space is a Polish space X endowed with a continuous action of G . A Borel G -space is a standard Borel space X endowed with a Borel action of G . Given a Borel G -space X , there exists aPolish topology τ on X such that ( X, τ ) is a Polish G -space; see [BK96, Theorem 5.2.1].A standard Borel group is, simply, a group object in the category of standard Borel spaces [ML98, Section III.6].Explicitly, a standard Borel group is a standard Borel space G that is also a group, and such that the groupoperation on G and the function G → G , x x − are Borel; see [Kec95, Definition 12.23]. Clearly, every Polishgroup is, in particular, a standard Borel group.The notion of Polish topometric space was introduced and studied in [BYM15, BYBM13, BY08, BYU10]. A topometric space is a Hausdorff space X endowed with a topology τ and a [0 , ∞ ]-valued metric d such that:(1) the metric-topology is finer than τ ; EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 5 (2) the metric is lower-semicontinuous with respect to τ , i.e. for every r ≥ { ( a, b ) ∈ X × X : d ( a, b ) ≤ r } is τ -closed in X × X .A Polish topometric space is a topometric space such that the topology τ is Polish and the metric is complete. A Polish topometric group is a Polish topometric space (
G, τ, d ) that is also a group, and such that G endowed withthe topology τ is a Polish group, and the metric d on G is bi-invariant1.2. Polish categories.
By definition, we let a Polish category be a category C enriched over the category of Polishspaces (regarded as a monoidal category with respect to binary products). Thus, for each pair of objects a, b of C , C ( a, b ) is a Polish space, such that for objects a, b, c , the composition operation C ( b, c ) × C ( a, b ) → C ( a, c ) iscontinuous.Suppose that C is a Polish category. For objects a, b of C , define Iso C ( a, b ) ⊆ C ( a, b ) be the set of C -isomorphisms a → b . While Iso C ( a, b ) is not necessarily a G δ subset of C ( a, b ), and hence not necessarily a Polish space whenendowed with the subspace topology, Iso C ( a, b ) is endowed with a canonical Polish topology, defined as follows. Fora net ( α i ) in Iso C ( a, b ) and α ∈ Iso C ( a, b ), set α i → α if and only if α i → α in C ( a, b ) and α − i → α − in C ( b, a ).One can then easily show the following. Lemma 1.2.
Adopt the notations above. Then
Iso C ( a, b ) is a Polish space. It is clear from the definition that, for every object a of C , Aut C ( a ) := Iso C ( a, a ) is a Polish group. Furthermore,the canonical (right and left) actions of Aut C ( a ) and Aut C ( b ) on C ( a, b ) are continuous. Definition 1.3.
Suppose that C and D are Polish categories, and F : C → D is a functor. We say that F iscontinuous if, for every pair of objects a, b of C , the map C ( a, b ) → D ( F ( a ) , F ( b )), f F ( f ) is continuous. Wesay that F is a topological equivalence if it is continuous, and there exists a continuous functor G : D → C such that GF is isomorphic to the identity functor I C , and F G is isomorphic to the identity functor I D .The notion of topological equivalence of categories is the natural analogue of the notion of equivalence of categoriesin the context of Polish categories; see [ML98, Section IV.4]. The same proof as [ML98, Section IV.4, Theorem 1]gives the following characterization of topological equivalences. Lemma 1.4.
Suppose that C and D are Polish categories, and F : C → D is a functor. The following assertionsare equivalent:(1) F is a topological equivalence;(2) each object of D is isomorphic to one of the form F ( a ) for some object a of C , and for each pair of objects c, d of C , the map C ( c, d ) → C ( F ( c ) , F ( d )) is a homeomorphism. Idealistic equivalence relations.
Suppose that C is a set. A σ -filter on C is a nonempty family F ofsubsets of C that is closed under countable intersections, and such that ∅ / ∈ F and if A ⊆ B ⊆ C and A ∈ F then B ∈ F . The dual notion is the one of σ -ideal. Thus, a nonempty family I of subsets of C is a σ -ideal if it is closedunder countable unions, C / ∈ I , and A ⊆ B ⊆ C and B ∈ I imply A ∈ I . Clearly, if F is a σ -filter on C , then { C \ A : A ∈ F} is a σ -ideal on C , and vice-versa. Thus, one can equivalently formulate notions in terms of σ -filtersor in terms of σ -ideals.If F is a σ -filter on C , then F can be thought of as a notion of “largeness” for subsets of C . Based on thisinterpretation, we use the “ σ -filter quantifier” notation “ F x , x ∈ A ” for a subset A ⊆ C to express the fact that A ∈ F . If P ( x ) is a unary relation for elements of C , “ F x , P ( x )” is the assertion that the set of x ∈ C that satisfy P ( x ) belongs to F . Example 1.5.
Suppose that C is a Polish space. A subset A of C is meager if it is contained in the union ofa countable family of closed nowhere dense sets. By the Baire Category Theorem [Kec95, Theorem 8.4], meagersubsets of C form a σ -ideal I C . The corresponding dual σ -filter is the σ -filter F C of comeager sets, which are thesubsets of C whose complement is meager.Suppose that X is a standard Borel space. We consider an equivalence relation E on X as a subset of X × X ,endowed with the product Borel structure. Consistently, we say that E is Borel or analytic, respectively, if it is MARTINO LUPINI a Borel or analytic subset of X × X . In the following, we will exclusively consider analytic equivalence relations,most of which will in fact be Borel. For an element x of X we let [ x ] E be its corresponding E -class.We now recall the notion of idealistic equivalence relation, initially considered in [Kec94]; see also [Gao09,Definition 5.4.9] and [KM16]. We will consider a slightly more generous definition than the one from [Kec94,Gao09, KM16]. The more restrictive notion is recovered as a particular case by insisting that the function s inDefinition 1.6 be the identity function of X . In the following definition, for a subset A of a product space X × Y and x ∈ X , we let A x = { y ∈ Y : ( x, y ) ∈ A } be the corresponding vertical section . Definition 1.6.
An equivalence relation E on a standard Borel space X is idealistic if there exist a Borel function s : X → X satisfying s ( x ) Ex for every x ∈ X , and a function C
7→ F C that assigns to each E -class C a σ -filter F C of subsets of C such that, for every Borel subset A of X × X , the set A s, F := (cid:8) x ∈ X : F [ x ] E x ′ , ( s ( x ) , x ′ ) ∈ A (cid:9) = (cid:8) x ∈ X : A s ( x ) ∈ F [ x ] E (cid:9) .is Borel.Idealistic equivalence relations arise naturally as orbit equivalence relations of Polish group actions. Supposethat G is a Polish group and X is a Polish G -space. Let E XG be the corresponding orbit equivalence relation on X , obtained by setting, for x, y ∈ X , xE XG y if and only if there exists g ∈ G such that g · x = y . Then E XG isan idealistic equivalence relation, as witnessed by the identity function s on X and the function C
7→ F C where A ∈ F C if and only if F G g , g · x ∈ A . (As in Example 1.5, F G denotes the σ -filter of comeager subsets of G .) Inparticular, if G is a Polish group, and H is a Polishable subgroup of G , then the coset equivalence relation E GH of H in G is Borel and idealistic.Suppose that E is an equivalence relation on a standard Borel space X . A Borel selector for E is a Borel function s : X → X such that, for x, y ∈ x , xEy if and only if s ( x ) = s ( y ). If E has a Borel selector, then E is Borel andidealistic; see [Gao09, Theorem 5.4.11]. (Precisely, an equivalence relation has a Borel selector if and only if it isBorel, idealistic, and smooth [Gao09, Definition 5.4.1].)1.4. Definable sets.
Definable sets are a generalization of standard Borel sets, and can be thought of as setsexplicitly presented as the quotient of a standard Borel space by a “well-behaved” equivalence relation E . Definition 1.7. A definable set X is a pair ( ˆ X, E ) where ˆ X is a standard Borel space and E is a Borel and idealisticequivalence relation on ˆ X . We think of ( ˆ X, E ) as a presentation of the quotient set X = ˆ X/E . Consistently, wealso write the definable set ( ˆ
X, E ) as ˆ
X/E . A subset Z of X is Borel if there is an E -invariant Borel subset ˆ Z ofˆ X such that Z = ˆ Z/E .We now define the notion of morphism between definable sets. Let X = ˆ X/E and Y = ˆ Y /F be definable sets.A lift of a function f : X → Y is a function ˆ f : ˆ X → ˆ Y such that f ([ x ] E ) = [ ˆ f ( x )] F for every x ∈ ˆ X . Definition 1.8.
Let X and Y be definable sets. A function f : X → Y is Borel-definable if it has a lift ˆ f : ˆ X → ˆ Y that is a Borel function. Remark 1.9.
Since Borel-definability is the only notion of definability we will consider in this paper, we willabbreviate “Borel-definable” to “definable”.We consider definable sets as objects of a category
DSet , whose morphisms are the definable functions. Weregard a standard Borel space X as a particular instance of definable set X = ˆ X/E where X = ˆ X and E is therelation of equality on X . This renders the category of standard Borel spaces a full subcategory of the category ofdefinable sets.If X = ˆ X/E and Y = ˆ Y /F are definable sets, then their product X × Y in DSet is the definable set X × Y :=( ˆ X × ˆ Y ) /( E × F ) , E × F being the equivalence relation on ˆ X × ˆ Y defined by setting ( x, y ) ( E × F ) ( x ′ , y ′ ) if andonly if xEx ′ and yF y ′ . (It is easy to see that E × F is Borel and idealistic if both E and F are Borel and idealistic.)Many of the good properties of standard Borel spaces, including all the ones listed in Proposition 1.1, generalizeto definable sets. Proposition 1.10.
Let as above
DSet be the category that has definable as objects and definable functions asmorphisms.
EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 7 (1) If X is a definable and A ⊆ X is a Borel subset, then A is itself a definable set;(2) If X, Y are definable sets, f : X → Y is an injective definable function, and A ⊆ X a Borel subset, then f ( A ) is a Borel subset of Y ;(3) If X, Y are definable sets, and f : X → Y is a bijective definable function, then the inverse function f − : Y → X is definable;(4) If X, Y are definable sets, and there exist injective definable functions f : X → Y and g : Y → X , thenthere exists a definable bijection h : X → Y ;(5) The category DSet has finite products, finite coproducts, equalizers, and pullbacks;(6) A definable function is monic in
DSet if and only if it is injective, and epic in
DSet if and only if it issurjective;(7) An inductive sequence of definable sets and injective definable functions has a colimit in
DSet .Proof. (1) is immediate from the definition. (2) and (3) are consequences of [KM16, Lemma 3.7], after observingthat the same proof there applies in the case of the more generous notion of idealistic equivalence relation consideredhere. (4) is a consequence of (2) and [MR12, Proposition 2.3]. Finally, (5), (6), and (7) are easily verified. (cid:3)
Occasionally we will need to consider quotients X = ˆ X/E where ˆ X is a standard Borel space E is an analytic equivalence relation on ˆ X that is not Borel and idealistic, or has not yet been shown to be Borel and idealistic. Inthis case, we say that X = ˆ X/E is a semidefinable set . Clearly, every definable set is, in particular, a semidefinableset. The notion of Borel subset and definable function are the same as in the case of definable sets. Thus, if X = ˆ X/E and Y = ˆ Y /F are semidefinable sets, Z ⊆ X is a subset and f : X → Y is a function, then f is definable if it has a Borel lift ˆ f : ˆ X → ˆ Y , and Z is Borel if there is a Borel E -invariant subset ˆ Z of ˆ X such that Z = ˆ Z/E .The category
SemiDSet has semidefinable sets as objects and definable functions as morphisms. Notice that, inparticular, an isomorphism from X to Y in SemiDSet is a bijection f : X → Y such that both f and the inversefunction f − : Y → X are definable. Lemma 1.11.
Suppose that X = ˆ X/E is a definable set, Y = ˆ Y /F is a semidefinable set. If X and Y areisomorphic in SemiDSet , then Y is a definable set.Proof. Suppose that the Borel function s X : ˆ X → ˆ X and the assignment C
7→ E C witness that E is idealistic. Byassumption, there exists a bijection f : X → Y such that f has a Borel lift α : ˆ X → ˆ Y , and f − has a Borel lift β : ˆ Y → ˆ X . For y, y ′ ∈ ˆ Y we have that yF y ′ if and only if β ( y ) Eβ ( y ′ ), whence F is Borel. We now show that F isidealistic.Define an assignment D
7→ F D from F -classes to σ -filters, by setting S ∈ F D if and only if α − ( S ) ∈ E C where f ( C ) = D . Consider also the Borel map s Y := α ◦ s X ◦ β : ˆ Y → ˆ Y . Then it is easy to verify that s Y and theassignment D
7→ F D witness that F is idealistic. (cid:3) Lemma 1.12.
Suppose that X = ˆ X/E and Y = ˆ Y /F are semidefinable sets. Assume that there exists a definablebijection f : X → Y (which is not necessarily an isomorphism in SemiDSet ). If E is Borel, then F is Borel.Proof. By assumption E ⊆ ˆ X × ˆ X is Borel, and F ⊆ ˆ Y × ˆ Y is analytic. Furthermore, f has a Borel lift ˆ f : ˆ X → ˆ Y .Since f is a bijection, we have that, for y, y ′ ∈ ˆ Y , yF y ′ ⇔ ∀ x, x ′ ∈ ˆ X , (cid:16) ( ˆ f ( x ) F y ∧ ˆ f ( x ′ ) F y ′ ) → xEx ′ (cid:17) .This shows that F is co-analytic. As F is also analytic, we have that F is Borel. (cid:3) Lemma 3.7 in [KM16] can be stated as the following proposition, which generalizes items (2) and (3) in Proposition1.10.
Proposition 1.13 (Kechris–Macdonald) . Let X = ˆ X/E be a definable set, Y = ˆ Y /F be semidefinable set suchthat F is Borel, and f : X → Y be a definable function. If f is injective, then the range of f a Borel subset of Y .If f is bijective, then the inverse function f − : Y → X is definable. The following result is a consequence of Lemma 1.12 and Proposition 1.13.
Corollary 1.14.
Suppose that X = ˆ X/E is a definable set, and Y = ˆ Y /F is a semidefinable set. If f : X → Y isa definable bijection, then Y is a definable set and f is an isomorphism in DSet . MARTINO LUPINI
Proof.
By Lemma 1.12, F is Borel. Whence, by Proposition 1.13, f is an isomorphism in SemiDSet . Since X isa definable set, it follows from Lemma 1.11 that Y is also a definable set, and f is an isomorphism in DSet . (cid:3) Definable groups.
A definable group can be simply defined as a group in the category
DSet in the senseof [ML98, Section III.6]. Thus, a definable group is a definable set G = ˆ G/E that is also a group, and such thatthe group operation G × G → G is definable, and the function G → G , x x − is also definable. As in the caseof sets, we regard a standard Borel group as a particular instance of definable group G = ˆ G/E where G = ˆ G is astandard Borel group and E is the relation of equality on ˆ G . Thus, standard Borel groups form a full subcategoryof the category of definable groups.Naturally, a semidefinable group will be a group in SemiDSets , i.e. a semidefinable set G = ˆ G/E that is alsoa group, and such that the group operation G × G → G is definable, and the function G → G that maps everyelement to its inverse is definable. Lemma 1.15. If G = ˆ G/E is a semidefinable group, then the equivalence relation E is Borel if and only if theidentity element of G , which is the E -class [ ∗ ] E of some element ∗ of ˆ G , is a Borel subset of ˆ G .Proof. Clearly, if E is Borel, then [ ∗ ] E is Borel. Conversely, suppose that [ ∗ ] E is Borel. If m : ˆ G × ˆ G → ˆ G and ζ : ˆ G → ˆ G are Borel lifts of the group operation in X and of the function that maps each element to its inverse,respectively, then we have that xEy if and only if m ( x, ζ ( y )) ∈ [ ∗ ] E . This shows that E is Borel. (cid:3) Corollary 1.16.
Suppose that G = ˆ G/E is a semidefinable group. If E is the orbit equivalence relation of a Borelaction of a Polish group H on the standard Borel space ˆ G , then G is a definable group.Proof. By [BK96, Theorem 5.2.1] one can assume that ˆ G is a Polish H -space, and E is the orbit equivalence relationof a continuous H -action on ˆ G . By [Gao09, Proposition 3.1.10], every E -class is Borel. Therefore E is Borel byLemma 1.15. Furthermore, E is idealistic by [Gao09, Proposition 5.4.10]. (cid:3) Remark 1.17.
A particular instance of definable group is obtained as follows. Suppose that G is a Polish groupand H is a Borel Polishable subgroup. Let E GH be the coset equivalence relation of H in G . The quotient group G/H is the quotient of G by the equivalence relation E GH . Since H is Polishable, E GH is the orbit equivalence relationof a Borel action of a Polish group on G . Thus, G/H = G/E GH is a definable group by Corollary 1.16. The definablegroups obtained in this way are called groups with a Polish cover in [BLP20].2. Strict C*-algebras
In this section we introduce the notion of strict Banach space and strict C*-algebra and some of their properties.Briefly, a strict Banach space is a Banach space whose unit ball is endowed with a Polish topology (called the stricttopology) that is coarser than the norm-topology and induced by a sequence of bounded seminorms. A suitablesemicontinuity requirement relates the norm and the strict topology. A strict C*-algebra is a strict Banach spacethat is also a C*-algebra with some suitable continuity requirement relating the C*-algebra operations and the stricttopology. The name is inspired by the strict topology on the multiplier algebra of a separable C*-algebra, whichwill be one of the main examples. Other examples are Paschke dual algebras of separable C*-algebras.2.1.
Strict Banach spaces.
Let X be a Banach space. We denote by Ball ( X ) its unit ball. A seminorm p on X is bounded if k p k := sup x ∈ Ball( X ) p ( x ) < ∞ . We say that p is contractive if k p k ≤ Definition 2.1.
A strict Banach space is a Banach space X such that Ball ( X ) is endowed with a topology (calledthe strict topology) such that, for some sequence ( p n ) of contractive seminorms on X , letting d be the pseudometricon Ball ( X ) defined by d ( x, y ) = X n ∈ ω − n p n ( x − y ) ,one has that:(1) d is a complete metric that induces the strict topology on Ball ( X );(2) Ball ( X ) contains a countable strictly dense subset;(3) k x k = sup n ∈ ω p n ( x ) for every x ∈ X . EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 9
Example 2.2.
Suppose that X is a separable Banach space. Then X is a strict Banach space where the stricttopology on Ball ( X ) is the norm topology. Example 2.3.
Suppose that Y is a separable Banach space, and Y ∗ is its Banach space dual. Then Y ∗ is a strictBanach space where the strict topology on Ball ( Y ∗ ) is the weak*-topology.Let X be a seminormed space, and consider the cone S ( X ) of bounded seminorms on X as a complete metricspace, with respect to the metric defined by d ( p, q ) = sup x ∈ Ball( X ) | p ( x ) − q ( x ) | . For a subset S ⊆ S ( X ), we let σ ( X, S ) be the topology on Ball ( X ) generated by S . We denote by Ball( S ) the set of contractive seminorms in S . If p ∈ S ( X ), S ⊆ S ( X ), and ( x n ) is a sequence in Ball ( X ), then we say that: • ( x n ) n ∈ ω is p -Cauchy if for every ε > n ∈ ω such that, for n, m ≥ n , p ( x n − x m ) < ε ; • ( x n ) n ∈ ω is S -Cauchy if it is p -Cauchy for every p ∈ S ; • Ball ( X ) is S -complete if, for every sequence ( x n ) n ∈ ω in Ball( X ), if ( x n ) n ∈ ω is S -Cauchy, then ( x n ) n ∈ ω is σ ( X, S )-convergent to some element of Ball ( X ).The following lemma is elementary. Lemma 2.4.
Suppose that X is a seminormed space. Let τ be a topology on Ball ( X ) . Assume that S and T aretwo sets of bounded seminorms on X such that the topologies σ ( X, S ) and σ ( X, T ) on Ball ( X ) coincide with τ .Then, for a sequence ( x n ) in Ball ( X ) , ( x n ) is S -Cauchy if and only if it is T -Cauchy. In this case, we say that ( x n ) is τ -Cauchy. It follows that Ball ( X ) is S -complete if and only if it is T -complete. In this case, we say that Ball ( X ) is τ -complete. In view of Lemma 2.4 one can equivalently define a strict Banach space as follows.
Definition 2.5.
A strict Banach space is a Banach space X such that Ball ( X ) is endowed with a topology (calledthe strict topology) such that, for some separable cone S of bounded seminorms on X , one has that:(1) the strict topology on Ball ( X ) is the σ ( X , S )-topology, and Ball ( X ) is strictly complete;(2) Ball ( X ) contains a countable strictly dense subset;(3) k x k = sup p ∈ Ball( S ) p ( x ) for every x ∈ X . Proposition 2.6.
Suppose that X is a strict Banach space. Then Ball ( X ) is a Polish topometric space whenendowed with the strict topology and the norm-distance.Proof. By definition, the strict topology on Ball ( X ) is Polish. Since the strict topology is induced by bounded semi-norms on X , it is coarser than the norm topology. The function ( x, y )
7→ k x − y k is strictly lower-semicontinuous,being the supremum of strictly continuous functions. Since the norm on X is complete, the distance ( x, y )
7→ k x − y k on Ball( X ) is complete. (cid:3) Suppose that X is a strict Banach space. We extend the strict topology of Ball( X ) to any bounded subset of X by declaring the function n Ball ( X ) → Ball ( X ) , z n z to be a homeomorphism with respect to the strict topology, where n Ball ( X ) = { z ∈ X : k z k ≤ n } .Then we have that addition and scalar multiplication on X are strictly continuous on bounded sets , and the normis strictly lower-semicontinuous on bounded sets . In particular n Ball ( X ) is a strictly closed subspace of m Ball ( X )for n ≤ m . Notice that, if Y is a norm-closed subspace of X such that Ball ( Y ) is strictly closed in Ball( X ), then Y is a strict Banach space with the induced norm and the induced strict topology. Definition 2.7.
Let X be a strict Banach space. The (standard) Borel structure on X is defined by declaring asubset A of X to be Borel if and only if A ∩ n Ball ( X ) is Borel for every n ≥ X is standard, as X is Borel isomorphic to the disjoint union of the standardBorel spaces ( n + 1) Ball ( X ) \ n Ball ( X ) for n ≥ Definition 2.8. If X and Y are strict Banach spaces. A bounded linear map T : X → Y is contractive if k T k ≤ strict if it is strictly continuous on bounded sets. A bounded seminorm p on X is strict if it is strictly continuouson bounded sets.Clearly, strict Banach spaces form a category where the morphisms are the strict contractive linear maps. Noticethat, if T is a strict, bijective, and isometric linear map T : X → Y between strict Banach spaces, then the inverse T − : Y → X is not necessarily strict, whence T is not necessarily an isomorphism in the category of strict Banachspaces. Nonetheless, T : X → Y is a Borel isomorphism , as both X and Y are standard Borel spaces. Definition 2.9.
Let X be a strict Banach space. Define S strict ( X ) to be the space of bounded, strict seminormson X .Notice that S strict ( X ) is a closed subspace of the complete metric space S ( X ). A sequence ( x n ) in Ball ( X ) isstrictly convergent if and only if it is S strict ( X )-Cauchy. A bounded linear map T : X → Y is strict if and only if p ◦ T ∈ S strict ( X ) for every p ∈ S strict ( Y ). Remark 2.10.
Suppose that X is a strict Banach space, and S is a separable cone of bounded, strict seminormson X that induces the strict topology on Ball ( X ). One can consider the globally defined topology σ ( X , S ) on X ,induced by all the seminorms in S . This topology coincides with the strict topology on Ball ( X ). However, it isnot first countable on the whole of X , unless X is a separable Banach space and the strict topology is equal to thenorm topology. Indeed, if the σ ( X , S )-topology on X is first-countable, then ( X , σ ( X , S )) is a Frechet space. Bythe Open Mapping Theorem for Frechet spaces [RR64, Theorem 8, page 120], any two comparable Frechet spacetopologies must be equal. Thus, σ ( X , S ) equals the norm topology. In particular, the norm-topology on Ball ( X )is equal to the strict topology, and it has a countable dense subset. Hence the norm-topology on X is separable.For future reference, we record the easily proved observation that a uniform limit of strictly continuous functionsis strictly continuous. Lemma 2.11.
Suppose that X and Y are strict Banach spaces, and A ⊆ Ball ( X ) . Suppose that f : A → Y is anfunction. Assume that there exists an sequence ( f n ) of strictly continuous function f n : A → Y such that lim n →∞ sup x ∈ A k f n ( x ) − f ( x ) k = 0 . Then f is strictly continuous. A standard Baire Category argument shows that one can characterize bounded subsets in terms of bounded,strict seminorms, as follows.
Lemma 2.12.
Let X be a strict Banach space. If A ⊆ X , then A is bounded if and only if, for every p ∈ S strict ( X ) , p ( A ) is a bounded subset of R . A natural way to obtain strict Banach spaces is via pairings.
Definition 2.13.
A Banach pairing is a bounded bilinear map h· , ·i : X × Y → Z , where X , Y, Z are Banach spaces.Define the σ ( X , Y )-topology to be the topology on X generated by the cone S Y of bounded seminorms x
7→ kh x, y ik for y ∈ Y .The following lemma is an immediate consequence of the definition of strict Banach space. Lemma 2.14.
Suppose that h· , ·i : X × Y → Z is a Banach pairing. Assume that: • Y, Z are norm-separable Banach spaces; • for every x ∈ X , k x k = sup y ∈ Ball( Y ) kh x , y ik ; • Ball ( X ) is σ ( X , Y ) -complete; • Ball ( X ) has a countable σ ( X , Y ) -dense subset.Then X is a strict Banach space where the strict topology on Ball ( X ) is the σ ( X , Y ) -topology. EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 11
Suppose that X is a norm-separable Banach space, and Y is a strict Banach spaces. A linear map T : X → Y is bounded if it maps bounded sets to bounded sets or, equivalently, k T k = sup x ∈ Ball( X ) k T ( x ) k < ∞ .This defines a norm on the space L ( X, Y ) of bounded linear maps X → Y . We also define the strict topology onBall( L ( X, Y )) to be the topology of pointwise convergence in the strict topology of Ball( Y ). Then one can easilyshow the following. Proposition 2.15.
Suppose that X is a norm-separable Banach space, and Y is a strict Banach space. Then L ( X, Y ) is a strict Banach space. Strict C*-algebras.
We now introduce the notion of strict C*-algebra. Given a C*-algebra A , we let A sa be the set of its self-adjoint elements. We also denote by M n ( A ) the C*-algebra of n × n matrices over A , whichcan be identified with the tensor product M n ( C ) ⊗ A . We refer to [Bla06, Dav96, Mur90, Ped79] for fundamentalnotions and results from the theory of C*-algebras. Definition 2.16. A strict C*-algebra is a C*-algebra A such that, for every n ≥ M n ( A ) is also a strict Banachspace satisfying the following properties:(1) the *-operation and the multiplication operation on M n ( A ) are strictly continuous on bounded sets;(2) the strict topology on Ball ( M n ( A )) is induced by the inclusionBall ( M n ( A )) ⊆ M n (Ball ( A )) ,where Ball ( A ) is endowed with the strict topology, and M n (Ball ( A )) is endowed with the product topology. Example 2.17.
Suppose that A is a separable C*-algebra. Then we have that A is a strict C*-algebra where, forevery n ≥
1, Ball ( M n ( A )) is endowed with the norm-topology.Suppose that A is a strict C*-algebra. Then, for every n ≥ M n ( A ) is also a strict C*-algebra. If A is a strictC*-algebra, then we regard A as a standard Borel space with respect to the standard Borel structure induced bythe strict topology on Ball ( A ) as in Definition 2.7. We say that a subset of A is Borel if it is Borel with respectto such a Borel structure. We have that the Borel structure on M n ( A ) (as a strict C*-algebra) coincides with theproduct Borel structure. Definition 2.18.
Suppose that A is a strict unital C*-algebra. A strict ideal of A is a norm-closed proper two-sidedBorel ideal J of A that is also a strict C*-algebra, and such that the inclusion map J → A is strict. Remark 2.19.
In order for J to be a strict ideal of A , we do not require that Ball ( J ) be strictly closed in Ball ( A )nor that the strict topology on Ball ( J ) be the subspace topology induced by the strict topology of Ball ( A ). Example 2.20.
Suppose that A is a strict unital C*-algebra and J ⊆ A is a norm-closed and norm-separableproper two-sided ideal of A . Then J is a strict ideal of A .We regard strict (unital) C*-algebras as objects of a category with strict (unital) *-homomorphisms as mor-phisms. (Recall that a bounded linear map is strict if it is strictly continuous on bounded sets.) If A ⊆ B , then wesay that A is strictly dense in B if Ball ( A ) is dense in Ball ( B ) with respect to the strict topology.It follows from the axioms of a strict C*-algebra that, if A is a strict C*-algebra, and p ( x , . . . , x n ) is a *-polynomial, then p defines a function A n → A that is strictly continuous on bounded sets. In particular, the setsof normal, self-adjoint, and positive elements of norm at most 1 are strictly closed in Ball ( A ). If f : [ − , n → n Ball ( C ) is a continuous function, then f induces by continuous functional calculus and Lemma 2.11 a strictlycontinuous functions ( x , . . . , x n ) f ( x , . . . , x n ) from the strictly closed set of n -tuples of pairwise commutingself-adjoint elements in Ball ( A ) to n Ball ( A ). Similarly, if f : Ball ( C ) n → k Ball ( C ) is a continuous function,then f induces by continuous functional calculus and Lemma 2.11 a strictly continuous function ( x , . . . , x n ) f ( x , . . . , x n ) from the strictly closed set of n -tuples of pairwise commuting normal elements in Ball ( A ) to k Ball ( A ).Suppose that A is a strict C*-algebra. Let Normal ( A ) be the Borel set of normal elements of A . For a ∈ Normal ( A ), the spectrum σ ( a ) is a closed subset of C . We consider the space Closed ( C ) of closed subsets of C as a standard Borel space endowed with the Effros Borel structure [Kec95, Section 12.C]. If X is a standard Borelspace and B is a basis of open subsets of C , then a function Φ : X → Closed ( C ) is Borel if and only if, for every U ∈ B , { x ∈ X : Φ( x ) ∩ U = ∅ } is Borel. The proof of the following lemma is standard; see [Sim95, Lemma 1.6]. Lemma 2.21.
Suppose that A is a strict C*-algebra. The function Normal ( A ) → Closed ( C ) , a σ ( a ) is Borel.Proof. It suffices to show that the map Normal ( A ) ∩ Ball ( A ) → Closed ( C ), a σ ( a ) is Borel. Observe that C hasa basis of open sets of the form U f := { x ∈ C : f ( x ) > } where f : C → [0 ,
1] is a continuous function. For such acontinuous function f : C → [0 , { a ∈ Normal ( A ) ∩ Ball ( A ) : σ ( a ) ∩ U f = ∅ } = { a ∈ Normal ( A ) ∩ Ball ( A ) : f ( a ) = 0 } ,which is closed in Normal ( A ) ∩ Ball ( A ). This concludes the proof. (cid:3) Suppose that A is a strict C*-algebra. Fix r ∈ (0 ,
1) and consider the set X = { x ∈ A : k − x k ≤ r } ⊆ A ) .Then, for x ∈ X we have that x is invertible, (cid:13)(cid:13) x − (cid:13)(cid:13) ≤ − r , and x − = X n ∈ ω x n .It follows from Lemma 2.11 that the function X → − r Ball ( A ), x x − is strictly continuous.More generally, suppose that Ω is an open subset of C , and f : Ω → C is a holomorphic function. Suppose that0 ∈ Ω and r > { z ∈ C : | z | ≤ r } ⊆ Ω. Then f admits a Taylor expansion f ( z ) = ∞ X n =0 a n z n that converges uniformly for | z | ≤ r [Ahl78, Chapter 5, Theorem 3 and Chapter 2, Theorem 2]. Fix b ∈ A and set X := { x ∈ A : k x − b k ≤ r } ⊆ (1 + k b k ) Ball ( A )Then for x ∈ X , f ( x − b ) := ∞ X n =0 a n ( x − b ) n ∈ A ;see [Ped89, Lemma 4.1.11]. Furthermore, the function X → c Ball ( A ), x f ( x − b ) is strictly continuous on X by Lemma 2.11, where c = sup {| f ( z ) | : | z | ≤ r } .2.3. Multiplier algebras.
Suppose that A is a separable C*-algebra. A double centralizer for A is a pair ( L, R )of bounded linear maps
L, R : A → A such that k L k = k R k and L ( x ) y = xR ( y ) for every x, y ∈ A . Let M ( A ) bethe set of double centralizers for A . Then M ( A ) is a C*-algebra with respect to the operations( L , R ) + ( L , R ) = ( L + L , R + R )( L , R ) ( L , R ) = ( L L , R R ) λ ( L, R ) = ( λL, λR )( L, R ) ∗ = ( R ∗ , L ∗ )and the norm k ( L, R ) k = k L k = k R k for ( L, R ) , ( L , R ) , ( L , R ) ∈ M ( A ) and λ ∈ C . The strict topology on Ball ( M ( A )) is the topology of pointwiseconvergence, namely the topology induced by the seminorms p a : ( L, R ) max {k L ( a ) k , k R ( a ) k} for a ∈ A .An element a ∈ A can be identified with the multiplier ( L a , R a ) ∈ M ( A ) defined by setting L a ( x ) = ax and R a ( x ) = xa for x ∈ X . This allows one to regard A as an essential ideal of M ( A ). (An ideal J of a C*-algebra B is essential if J ⊥ := { b ∈ B : bJ = 0 } is zero or, equivalently, J has nonzero intersection with every nonzero ideal of B .) If ( v n ) n ∈ ω is an approximate unit for A [HR00, Definition 1.7.1] then, by definition, ( v n ) strictly converges to1 in Ball ( M ( A )). In particular, Ball ( A ) is strictly dense in Ball ( M ( A )). EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 13
If ( x i ) i ∈ ω is a strictly Cauchy sequence in Ball ( M ( A )), in the sense that ( x i ) i ∈ ω is p a -Cauchy for every a ∈ A ,then setting L ( a ) := lim i →∞ x i aR ( a ) := lim i →∞ ax i for a ∈ A defines a double centralizer ( L, R ) ∈ Ball ( M ( A )) that is the strict limit of ( x i ) i ∈ ω in Ball ( M ( A )). For n ≥
1, one can identify M n ( M ( A )) with M ( M n ( A )) and consider the corresponding strict topology. From theabove remarks and Lemma 2.14, one easily obtains the following; see [Far19, Chapter 13] or [WO93, Chapter 2]. Proposition 2.22.
Let A be a separable C*-algebra. Then M ( A ) is a strict unital C*-algebra containing A as astrictly dense essential strict ideal where, for every n ≥ , the strict topology on Ball ( M ( A )) is as described above,and M n ( M ( A )) is identified with M ( M n ( A )) . Example 2.23.
When A is the algebra K ( H ) of compact operators on a separable Hilbert space, then M ( A ) = B ( H ) and the strict topology on Ball ( B ( H )) is the strong-* topology [Bla06, Proposition I.8.6.3]. Example 2.24.
One can also regard B ( H ) as the dual space of the Banach space L ( H ) of trace-class operators.This turns B ( H ) into a strict Banach space, where the strict topology on Ball ( B ( H )) is the weak* topology,which coincides with the weak operator topology [Bla06, Definition I.8.6.2]. As the identity map Ball ( B ( H )) → Ball ( B ( H )) is strong-*–weak continuous, the strong-* topology and weak operator topology on Ball ( B ( H )) definethe same standard Borel structure on B ( H ).One can define as above the strict topology on the whole multiplier algebra M ( A ) to be the topology of pointwiseconvergence of double multipliers. However, this topology on M ( A ) is not first countable whenever A is not unital;see Remark 2.10.Suppose that A is a separable C*-algebra, and X is a compact metrizable space. One can then consider the separa-ble C*-algebra C ( X, A ) of continuous functions X → A . Let also C β ( X, M ( A )) be the C*-algebra of strictly contin-uous bounded functions X → M ( A ). There is an obvious unital *-homomorphism C β ( X, M ( A )) → M ( C ( X, A )),where C β ( X, M ( A )) acts on C ( X, A ) by pointwise multiplication. The unital *-homomorphism C β ( X, M ( A )) → M ( C ( X, A )) is in fact a *-isomorphism [APT73, Corollary 3.4]. We can thus identify C β ( X, M ( A )) with M ( C ( X, A ))and regard it as a strict C*-algebra. Observe that, for t ∈ X , the function Ball ( C β ( X, M ( A ))) → Ball ( M ( A )), f f ( t ) is strictly continuous. We let C ( X, M ( A )) be the C*-algebra of norm-continuous functions X → M ( A ),which is a C*-subalgebra of C β ( X, M ( A )). Lemma 2.25.
Suppose that A is a separable C*-algebra, and X is a compact metrizable space. Then C ( X, M ( A )) is a Borel subset of C β ( X, M ( A )) .Proof. Fix a compatible metric d on X , and a countable dense subset X of X . Clearly, it suffices to show thatBall ( C ( X, M ( A ))) is a Borel subset of Ball ( C β ( X, M ( A ))). Fix, for every k ∈ ω , a finite cover (cid:8) A k , . . . , A kℓ k − (cid:9) of X consisting of open sets of diameter less than 2 − k , and fix elements t ki ∈ A ki for i < ℓ k . We have that a strictlycontinuous function f : X → Ball ( M ( A )) is norm-continuous if and only if, for every n ∈ ω there exists k ∈ ω suchthat, for every i < ℓ k and s ∈ A ki , (cid:13)(cid:13) f ( s ) − f (cid:0) t ki (cid:1)(cid:13)(cid:13) ≤ − k . Since 2 − k Ball ( M ( A )) is strictly closed and f is strictlycontinuous, we have that f is norm-continuous if and only if for every n ∈ ω there exists k ∈ ω such that, for every i < ℓ k and for every s ∈ A ki ∩ X , (cid:13)(cid:13) f ( s ) − f (cid:0) t ki (cid:1)(cid:13)(cid:13) ≤ − k . This shows that the set of norm-continuous functions isBorel. (cid:3) Corollary 2.26.
Suppose that A is a separable C*-algebra. Then the set C ([0 , , M ( A )) of norm-continuous paths [0 , → M ( A ) is a Borel subset of C β ([0 , , M ( A )) . Suppose that A and B are separable C*-algebra. A morphism from A to B in the sense of [Wor80, Wor91, WN92,Wor95] is a *-homomorphism ϕ : A → M ( B ) such that ϕ ( A ) B is norm-dense in B . (This is called S -morphismin [Val85, Definition 0.2.7] and a nondegenerate *-homomorphism in [Lan95].) We recall the well-known fact thatthere is a correspondence between morphisms from A to B and strict unital *-homomorphisms M ( A ) → M ( B );see [Lan95, Proposition 2.1]. Lemma 2.27.
Let A and B be separable C*-algebra. • Suppose that ψ : M ( A ) → M ( B ) is a strict unital *-homomorphism. Then ψ | A is a morphism from A to B . • Conversely, if ϕ is a morphism from A to B , then ϕ extends to a unique strict unital *-homomorphism ¯ ϕ : M ( A ) → M ( B ) . If ϕ is injective, then ¯ ϕ is injective. • If ( e n ) is an approximate unit for A , then a *-homomorphism ϕ : A → M ( B ) is a morphism from A to B if and only if ( ϕ ( e n )) strictly converges to . A further characterization of morphisms is provided in [Val85, Lemme 0.2.6] and [I´or80, Proposition 1.1]. Itfollows from Lemma 2.27 that the composition of morphisms A → B and B → C is meaningful, and it gives amorphism A → C .Suppose that A, B are separable C*-algebras. A *-homomorphism ϕ : A → M ( B ) is quasi-unital [JT91, Defini-tion 1.3.13] (also called strict [Lan95, page 49]) if there exists a projection p ϕ ∈ M ( B ), called the relative unit of ϕ , such that ϕ ( A ) B = p ϕ B . One has the following generalization of Lemma 2.27; see [Lan95, Corollary 5.7]. Lemma 2.28.
Let A and B be separable C*-algebra. • Suppose that ψ : M ( A ) → M ( B ) is a strict *-homomorphism. Then ψ | A is a quasi-unital *-homomorphismfrom A to M ( B ) with relative unit ψ (1) . • Conversely, if ϕ is a quasi-unital *-homomorphism from A to M ( B ) with relative unit p ϕ , then ϕ extendsto a unique strict *-homomorphism ¯ ϕ : M ( A ) → M ( B ) with ¯ ϕ = p ϕ . If ϕ is injective, then ¯ ϕ is injective. • If ( e n ) is an approximate unit for A , then a *-homomorphism ϕ : A → M ( B ) is quasi-unital if and only if ( ϕ ( e n )) is strictly Cauchy. We now observe that the category of multiplier algebras of separable C*-algebras, regarded as a full subcategory ofthe category of strict unital C*-algebras, can be regarded as a Polish category; see Section 1.2. This means that, forevery separable C*-algebras A and B , the set Mor( M ( A ) , M ( B )) of strict unital *-homomorphisms M ( A ) → M ( B )is a Polish space, and composition of morphisms is a continuous function.Following [Wor95] we consider Mor ( M ( A ) , M ( B )) as endowed with the topology of pointwise strict convergence.This is the subspace topology induced by regarding, as in Lemma 2.27, Mor ( M ( A ) , M ( B )) as a subspace ofBall ( L ( A, M ( B ))), where L ( A, M ( B )) is the space of bounded linear maps from A to M ( B ). (Recall that, if X isa Banach space and Y is a strict Banach space, then the space L ( X, Y ) of bounded linear maps X → Y is a strictBanach space when Ball ( L ( X, Y )) is endowed with the topology of pointwise strict convergence; see Proposition2.15.) As Mor ( M ( A ) , M ( B )) is a G δ subset of Ball ( L ( A, M ( B ))), it is a Polish space with the induced topology.It is easy to see that this turns the category of muliplier algebras of separable C*-algebras into a Polish category.If A, B are separable C*-algebras, then the space Iso ( M ( A ) , M ( B )) of isomorphisms M ( A ) → M ( B ) in thecategory of strict unital C*-algebras endowed with the Polish topology as in Lemma 1.2 can be identified, viathe correspondence given by Lemma 2.27, with the space Iso ( A, B ) of *-isomorphisms A → B endowed with thetopology of pointwise norm-convergence.Consider now the category of locally compact second countable Hausdorff spaces, where a morphism is simplya continuous map. Given locally compact second countable Hausdorff spaces X, Y , let Mor (
X, Y ) be the set of allcontinuous maps X → Y . This is endowed with a Polish topology called the compact-open topology, that has assubbasis of open sets the sets of the form( K, U ) := { f ∈ Mor (
X, Y ) : f ( K ) ⊆ U } for a compact subset K of X and an open subset U of Y . This turns the category of locally compact secondcountable Hausdorff spaces and continuous maps into a Polish category. We let Iso ( X, Y ) ⊆ Mor (
X, Y ) be the setof homeomorphisms X → Y . The Polish topology induced on Iso ( X, Y ) as in Lemma 1.2 was shown in [Are46,Theorem 5], where it is called the g -topology, to have as subbasis of open sets the sets of then ( K, Y \ L ) where K, L are closed sets and at least one between K and L is compact. For a locally compact second countable Hausdorff space X , let X + be its one-point compactification, obtained by adjoining a point at infinity ∞ X . Each f ∈ Iso (
X, Y )admits a unique extension to f + ∈ Iso( X + , Y + ) that fixes the point at infinity, in the sense that f + ( ∞ X ) = ∞ Y .By [Are46, Theorem 5], the assignment f f + defines a homeomorphism from Iso ( X, Y ) onto the closed subsetof Iso( X + , Y + ) consisting of the homeomorphisms that fix the point at infinity.Given a locally compact second countable Hausdorff space X , we let C ( X ) be the separable C*-algebra ofcontinuous complex-valued functions on X vanishing at infinity. Its multiplier algebra M ( C ( X )) is the algebra EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 15 C b ( X ) of bounded continuous complex-valued functions on X . The unit ball Ball ( C b ( X )) of C b ( X ) = M ( C ( X ))endowed with the strict topology can be identified with the space Mor ( X, Ball ( C )) of continuous functions X → Ball ( C ) endowed with the compact-open topology. Every separable commutative C*-algebra A is isomorphic to C ( ˆ A ), where ˆ A is the locally compact second countable Hausdorff space of nonzero homomorphisms A → C (thespectrum of A ).A continuous map f : X → Y induces a strict unital *-homomorphism C b ( Y ) → C b ( X ) given by ϕ f : C b ( Y ) → C b ( X ), a a ◦ f . This defines a fully faithful contravariant functor from the category of locally compact secondcountable Hausdorff spaces to the category of strict unital C*-algebras. In fact, the assignment Mor ( X, Y ) → Mor ( C b ( Y ) , C b ( X )), f ϕ f is a homeomorphism, where Mor( X, Y ) is endowed as above with the compact-opentopology and Mor ( C b ( X ) , C b ( Y )) is endowed with the topology of pointwise strict convergence. Thus, by Lemma1.4, the assignment X → C b ( X ) is a contravariant topological equivalence of categories from the Polish categoryof locally compact second countable Hausdorff spaces to the Polish category of multiplier algebras of commutative separable C*-algebras; see Definition 1.3.2.4. Essential commutants and Paschke dual algebras.
Suppose that B is a separable C*-algebra, and C ⊆ M ( B ) is a separable C*-subalgebra. Define then the essential commutant D ( C ) of C in M ( B ) to be the C*-algebra { x ∈ M ( B ) : ∀ c ∈ B, [ x, c ] ∈ B } ,where [ x, c ] is the commutator xc − cx . Define the strict topology on Ball( D ( C )) to be the topology generated bythe seminorms x max {k xb k , k bx k , k [ x, c ] k} for c ∈ C and b ∈ B . If ( v n ) n ∈ ω is a approximate unit for B that is approximately central for C [HR00, Definition3.2.4], then ( v n ) n ∈ ω converges strictly to 1 in Ball( D ( B )).We have that D ( C ) is strictly complete. Indeed, consider a strictly Cauchy sequence ( x i ) i ∈ ω in Ball( D ( C )).Then we have that ( x i ) i ∈ ω converges to some x ∈ Ball ( M ( B )) in the strict topology of M ( B ). For every c ∈ C ,the sequence ([ x i , c ]) i ∈ ω is norm-Cauchy in B , whence it norm-converges to some element of B , which must beequal to [ x, c ]. This shows that x ∈ Ball ( D ( C )) is the strict limit of ( x i ) i ∈ ω in Ball ( D ( C )). For n ≥
1, we canidentify M n ( D ( C )) with D (∆ n ( C )) ⊆ M ( M n ( B )), where ∆ n ( C ) ⊆ M n ( B ) is the image of C under the diagonalembedding ∆ n : B → M n ( B ). From the above remarks and Lemma 2.14 we thus obtain the following. Proposition 2.29.
Let B be a separable C*-algebra, and let C ⊆ M ( B ) be a separable C*-subalgebra. Let D ( C ) bethe corresponding essential commutant. Then D ( C ) is a strict C*-algebra containing B as a strictly dense essentialstrict ideal where, for every n ≥ , M n ( D ( C )) is identified with D (∆ n ( C )) , and Ball ( D (∆ n ( C ))) is endowed withthe strict topology described above. Suppose now as above that B is a separable C*-algebra, and C ⊆ M ( B ) is a separable C*-subalgebra. Let also I ⊆ C be a closed two-sided ideal. Define the essential annihilator D ( C//I ) = { x ∈ D ( C ) : ∀ a ∈ I, xa ∈ B } ,which is a closed two-sided ideal of D ( C ). The strict topology on Ball ( D ( C//I )) is the topology generated by theseminorms x max {k xb k , k bx k , k [ x, c ] k} for b ∈ B ∪ I and c ∈ C . A straightforward argument as above gives the following. Proposition 2.30.
Let B be a separable C*-algebra, let C ⊆ M ( B ) be a separable C*-subalgebra, and I ⊆ C be a closed two-sided ideal. Let D ( C ) be the corresponding essential commutant, and D ( C//I ) be the essentialannihilator. Then D ( C//I ) is a strict ideal of D ( C ) , where for every n ≥ , M n ( D ( C//I )) is identified with D (∆ n ( C ) // ∆ n ( I )) and Ball ( D (∆ n ( C ) // ∆ n ( I ))) is endowed with the strict topology described above. Example 2.31.
Suppose that A is a separable unital C*-algebra, J is a closed two-sided ideal of A , and ρ : A → B ( H ) is a nondegenerate representation of A that is ample , in the sense that ρ ( A ) ∩ K ( H ) = { } . We regard B ( H )as the multiplier algebra of K ( H ). The Paschke dual D ρ ( A ) as defined in [HR00, Definition 5.1.1] is the essentialcommutant D ( ρ ( A )) of ρ ( A ) inside B ( H ); see also [Pas81]. The relative dual algebra D ρ ( A//J ) as defined in[HR00, Definition 5.3.2] is the strict ideal D ( ρ ( A ) //ρ ( J )) of D ρ ( A ) = D ( ρ ( A )). Homotopy of projections.
Suppose that A is a strict unital C*-algebra. Recall that a strict ideal of A is aproper norm-closed Borel two-sided ideal J of A that is also a strict C*-algebra and such that the inclusion map J → A is a strict *-homomorphism. Definition 2.32.
A strict (unital) C*-pair is a pair ( A , J ) where A is a strict (unital) C*-algebra and J is a strictideal of A .We regard strict unital C*-pairs as objects of a category, where a morphism from ( A , I ) to ( B , J ) is a strict unital*-homomorphism ϕ : A → B that maps I to J .Every strict unital C*-pair ( A , J ) determines a quotient unital C*-algebra A / J . If A / I and B / J are two unitalC*-algebras obtained in this way, then we say that a unital *-homomorphism ϕ : A / I → B / J is definable if it hasa Borel lift (or a Borel representation in the terminology of [Far11, Gha15]). This is a Borel function f : A → B (which is not necessarily a *-homomorphism) such that ϕ ( a + I ) = f ( a )+ J for every a ∈ A . The notion of definableunital *-homomorphisms determines a category, whose objects are strict unital C*-pairs and whose morphisms arethe definable unital *-homomorphism. When the strict unital C*-pair ( A , J ) is considered as the object of thiscategory, we call it a unital C*-algebra with a strict cover, and denote it by A / J , as we think of it as a unitalC*-algebra explicitly presented as the quotient of a strict unital C*-algebra by a strict ideal. The category ofunital C*-algebras with a strict cover thus has unital C*-algebras with strict cover as objects and definable unital*-homomorphisms as morphisms. The notion of a unital C*-algebra with a strict cover is the analogue in the contextof C*-algebras to the notion of group with a Polish cover considered in [BLP20]; see Remark 1.17.Notice that every strict unital *-homomorphism ( A , I ) → ( B , J ) between strict unital C*-pairs induces a definableunital *-homomorphism A / J → B / J between the corresponding unital C*-algebras with a strict cover. This allowsone to regard the category of strict unital C*-pairs as a subcategory of the category of unital C*-algebras with astrict cover. These categories have the same objects, but different morphisms.If ( A , J ) is a strict unital C*-pair and a, b ∈ A , we write a ≡ b mod J if a − b ∈ J . If a ∈ M n ( A ) and b ∈ M k ( A ),then we set a ⊕ b = (cid:20) a b (cid:21) ∈ M n + k ( A ) .We let 1 n be the identity element of M n ( A ) and 0 n be the zero element of M n ( A ).Suppose that ( A , J ) is a strict unital C*-pair. A positive element of Ball( A ) is a projection mod J if p ≡ p mod J or, equivalently, p + J is a projection in A / J . Define the set Proj ( A / J ) ⊆ Ball ( A ) to be the Borel set of projectionsmod J in A . The Borel structure on Proj ( A / J ) is induced by the Polish topology defined by declaring a net ( p i ) i ∈ I to converge to p if and only if p i → p strictly in Ball( A ) and p i − p i → p − p strictly in 2Ball( J ). (Recall that thestrict topology on Ball ( J ) might be different from the topology induced by the strict topology on Ball ( A ).)We also say that an element u of Ball( A ) is a unitary mod J if uu ∗ ≡ J and u ∗ u ≡ J or, equivalently, u + J is a unitary in A / J . We let U ( A / J ) be the Borel set of unitaries mod J in A . The Borel structure on U ( A / J )is induced by the Polish topology defined by declaring a net ( u i ) i ∈ I to converge to u if and only if u i → u strictlyin Ball ( A ), u i u ∗ i − → uu ∗ − J ), and u ∗ i u i − → u ∗ u − J ).More generally, an element v of Ball ( A ) is called a partial unitary mod J if uu ∗ ≡ uu ∗ mod J and uu ∗ is amod J projection or, equivalently, if v + J is a partial unitary in A / J as in [RLL00, 8.2.12]. We let PU ( A / J ) bethe Borel set of mod J partial unitaries in A . In a similar fashion one can define the Borel set PI( A / J ) of mod J partial isometries in A , consisting of those v ∈ Ball ( A ) such that vv ∗ and v ∗ v are mod J projections.In the rest of this section we record some lemmas about unitaries and projections modulo a strict ideal in a strictunital C*-algebra. The content of these lemmas can be summarized as the assertion that a homotopy betweenprojections and unitaries in a unital C*-algebra with a strict cover is witnessed by unitary elements in the path-component of the identity of the unitary group that can be chosen in a Borel fashion. The proofs follow standardarguments from the literature on K-theory for C*-algebras; see [RLL00, HR00, Bla98, WO93].Given elements y , . . . , y n of Ball ( A ), subject to a certain relation P ( y , . . . , y n ), we say that an element z ∈ Ball ( A ) satisfying a relation R ( y , . . . , y n , z ) can be chosen in a Borel fashion (from y , . . . , y n ) if there isa Borel function ( y , . . . , y n ) z ( y , . . . , y ) that assign to each n -tuple ( y , . . . , y n ) in Ball ( A ) satisfying P anelement z ( y , . . . , y n ) in Ball ( A ) such that ( y , . . . , y n , z ( y , . . . , y )) satisfies R . In other words, the set of tuples( y , . . . , y n , z ) ∈ Ball ( A ) n × Ball ( A ) such that ( y , . . . , y n ) satisfies P and ( y , . . . , y n , z ) satisfies R has a Boreluniformization [Kec95, Section 18.A]. EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 17
Suppose that A is a unital C*-algebra. Let Proj( A ) be the set of projections in A . Two projections p, q in A are: • Murray–von Neumann equivalent if there exists v ∈ A such that v ∗ v = p and vv ∗ = q , in which case wewrite p ∼ MvN q ; • unitary equivalent if there exists u ∈ U ( A ) such that u ∗ qu = p ; • homotopic if there is a norm-continuous path ( p t ) t ∈ [0 , in Proj ( M n ( A )) with p = p and p = q . Lemma 2.33.
Suppose that ( A , J ) is a strict unital C*-pair, d ∈ N , and u ∈ d Ball ( A ) satisfies k u − J k ≤ / .Then one can choose in a Borel way a self-adjoint element y ∈ Ball ( A ) such that e iy ≡ u mod J .Proof. Consider u − J ∈ A / J , and observe that there exists a ∈ A such that k a k ≤ / a + J = u − J ,which can be chosen in a Borel way by strict continuity of the continuous functional calculus. Hence, setting˜ u := a + 1 ∈ ( d + 1) Ball ( A ), we have that ˜ u ≡ u mod J and k ˜ u − k ≤ /
2. Thus, after replacing d with d + 1 and u with ˜ u , we can assume that k u − k ≤ / D → C be an holomorphic branch of the logarithm defined on { z ∈ C : | z − | < } . Considering theholomorphic functional calculus, one can define the element log( u ) ∈ A . Aslog( z ) = ∞ X n =0 (1 − z ) n n is the uniformly convergent power series expansion in { z ∈ C : | z | ≤ / } , we have thatlog( u ) = ∞ X n =0 (1 − u ) n n .In particular, k log( u ) k ≤
1. Define y := log( u ) + log( u ) ∗ ∈ Ball ( A sa ) .Then we have that y ≡ log( u ) mod J satisfies exp ( iy ) ≡ u mod J . (cid:3) Corollary 2.34.
Suppose that ( A , J ) is a strict unital C*-pair, and u, w ∈ A are mod J unitaries. Then therefollowing assertions are equivalent:(1) there is a norm-continuous path from u + J to w + J in A / J ;(2) there exists ℓ ≥ and y , . . . , y ℓ ∈ Ball ( A sa ) such that e iy · · · e iy ℓ u ≡ w mod J . Lemma 2.35.
Suppose that ( A , J ) is a strict unital C*-pair, and p, q, x ∈ Ball ( A ) are such that p, q are mod J projections, x ∗ x ≡ p mod J , and xx ∗ ≡ q mod J . Then one can choose Y , . . . , Y ℓ ∈ Ball ( M ( A ) sa ) in a Borelfashion from p, q, x such that, setting U := e iY · · · e iY ℓ , one has that U ∗ ( q ⊕ U ≡ ( p ⊕
0) mod M ( J ) and ( q ⊕ U ( p ⊕ ≡ x ⊕ M ( J ) ,where ℓ ≥ does not depend on ( A , J ) and p, q, x .Proof. Consider the mod M ( J ) unitary X := (cid:20) x − q − p x ∗ (cid:21) ∈ M ( A ) .Notice that X satisfies X ∗ ( q ⊕ d ) X ≡ ( p ⊕ d ) mod M ( J )and ( q ⊕ d ) X ( p ⊕ d ) ≡ x ⊕ d mod M ( J ) .Consider the norm-continuous path of mod M ( J ) unitaries X t := (cid:20) cos (cid:0) πt (cid:1) x − (cid:0) − sin (cid:0) πt (cid:1)(cid:1) q (cid:0) − sin (cid:0) πt (cid:1)(cid:1) p − (cid:0) πt (cid:1) x ∗ (cid:21) for t ∈ [0 , X t ) t ∈ [0 , does not depend on ( A , J ) and ( p, q, x ). Fix ℓ ≥ t, s ∈ [0 ,
1] satisfy | s − t | ≤ /ℓ , then k X t − X s k ≤ / i ∈ { , , . . . , ℓ } we have that (cid:13)(cid:13) X i/ℓ − X ( i +1) /ℓ (cid:13)(cid:13) ≤ / Y ∈ Ball ( A sa ) such that e iY ≡ X /ℓ mod M ( J ). Thus R := exp ( iY ) − X /ℓ ∈ M ( J ) . Consider now X /ℓ and the fact that (cid:13)(cid:13) X /ℓ − X /ℓ (cid:13)(cid:13) ≤ / (cid:13)(cid:13) exp ( iY ) − (cid:0) X /ℓ + R (cid:1)(cid:13)(cid:13) ≤ / (cid:13)(cid:13) − exp ( − iY ) (cid:0) X /ℓ + R (cid:1)(cid:13)(cid:13) ≤ / Y ∈ Ball ( A sa ) such thatexp ( iY ) ≡ exp ( − iY ) (cid:0) X /ℓ + R (cid:1) ≡ exp ( − iY ) X /ℓ mod M ( J )and hence exp ( iY ) exp ( iY ) ≡ X /ℓ mod M ( J ) .Proceeding recursively in this way, one can choose Y , . . . , Y ℓ ∈ Ball ( A sa ) in a Borel fashion such thatexp ( iY ) · · · exp ( iY ℓ ) ≡ X mod M ( J ) .Then we have that, setting U := exp ( iY ) · · · exp ( iY ℓ ), U ∗ ( q ⊕ d ) U ≡ X ∗ ( q ⊕ d ) X ≡ p ⊕ d mod M ( J )and ( q ⊕ d ) U ( p ⊕ d ) ≡ ( q ⊕ d ) X ( p ⊕ d ) ≡ x ⊕ d mod M ( J ) .This concludes the proof. (cid:3) Lemma 2.36.
Suppose that ( A , J ) is a strict unital C*-pair, and p, q ∈ A sa are mod J projections such that k p − q k ≤ / . Then one can choose y , . . . , y ℓ ∈ Ball ( A sa ) in a Borel fashion from p, q such that, setting u := e iy · · · e iy n , one has that u ∗ qu ≡ p mod J , where ℓ ≥ does not depend on ( A , J ) and p, q .Proof. As in the proof of [RLL00, Proposition 2.2.4], consider the norm-continuous path of mod J projections a t := (1 − t ) p + tq for t ∈ [0 , K = [ − / , / ∪ [3 / , / ⊆ R , and f : K → C be the continuousfunction that is 0 on [ − / , /
4] and 1 on [3 / , / p t := f ( a t ) for t ∈ [0 ,
1] is a norm-continuous path ofmod J projections from p to q . Notice that the uniform continuity moduli of t a t and t p t do not depend on( A , J ) and p, q .Thus, there exists k ∈ ω (that does depend on ( A , J ) and p, q ) such that, for every t, s ∈ [0 ,
1] such that | t − s | ≤ /k , one has that k p t − p s k ≤ /
6. Thus, p = p, p /k , p /k , . . . , p = q are mod J projections (that dependin a Borel way from p, q by strict continuity of the continuous functional calculus) such that (cid:13)(cid:13) p ( i +1) /k − p i/k (cid:13)(cid:13) ≤ / i ∈ { , , . . . , k − } and ( p i (1 − s )+(1+ i ) sk ) s ∈ [0 , is a norm-continuous path from p i/k to p ( i +1) /k (whose modulus ofcontinuity does not depend on A and p, q ∈ A ) satisfying (cid:13)(cid:13)(cid:13) p i (1 − s )+(1+ i ) sk − p i/k (cid:13)(cid:13)(cid:13) ≤ / s ∈ [0 , k p t − p k ≤ / t ∈ [0 , x t := (2 p −
1) ( p t − p ) + 1 EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 19
By definition, we have that x = 1. Notice that x t ≡ pp t + ( p −
1) ( p t −
1) mod J px t ≡ pp t ≡ x t p t mod J p t x ∗ t x t ≡ x ∗ t px t ≡ x ∗ t x t p t mod J and px t x ∗ t ≡ x t p t x ∗ t ≡ x t x ∗ t p mod J .This implies that p | x ∗ t | ≡ | x ∗ t | p mod J and p t | x t | ≡ | x t | p t mod J for t ∈ [0 , k x t − k = k (2 p −
1) ( p t − p ) k ≤ k p − k k p t − p k ≤ / x t is invertible. Let x t := u t | x t | be its polar decomposition, where u t is a unitary. Then we have that pu t ≡ u t p t mod J . Indeed, pu t ≡ px t | x t | − ≡ x t p t | x t | − ≡ x t | x t | − p t ≡ u t p t mod J .Thus u ∗ t pu t ≡ p t mod J for t ∈ [0 ,
1] and in particular u ∗ pu ≡ q mod J .Notice that ( u t ) t ∈ [0 , is a norm-continuous path, whose modulus of continuity does not depend on ( A , J ) and p, q . Therefore, there exists k ≥ A , J ) and p, q ) such that, whenever s, t ∈ [0 ,
1] satisfy | s − t | ≤ /k , we have k u t − u s k ≤ /
2. By Lemma 2.33 one can then choose in a Borel way y , . . . , y k ∈ Ball ( A sa )such that, setting u := exp ( iy ) · · · exp ( iy k ), then u ≡ u mod J and hence u ∗ pu ≡ q mod J . This concludes theproof. (cid:3) Lemma 2.37.
Suppose that ( A , J ) is a strict unital C*-pair, and p, q ∈ A are mod J projections that satisfy pq ≡ qp ≡ J . Then one can choose Y , . . . , Y ℓ ∈ Ball ( M ( A ) sa ) in a Borel fashion from p, q such that,setting U := e iY · · · e iY ℓ , one has that U ∗ ( p ⊕ q ) U ≡ ( p + q ) ⊕ M ( J ) , where ℓ ≥ does not depend on ( A , J ) and p, q .Proof. Consider the path r t := (cid:20) (cid:21) p + (cid:20) cos (cid:0) πt (cid:1) cos (cid:0) πt (cid:1) sin (cid:0) πt (cid:1) cos (cid:0) πt (cid:1) sin (cid:0) πt (cid:1) sin (cid:0) πt (cid:1) (cid:21) q for t ∈ [0 , M ( J ) projections in M ( A ) from ( p + q ) ⊕ p ⊕ q , whosemodulus of continuity does not depend on A and p, q . Therefore, the conclusion follows from Lemma 2.36. (cid:3) Lemma 2.38.
Suppose that ( A , J ) is a strict unital C*-pair, and u, v ∈ A are mod J unitaries. Then one can choose y , . . . , y ℓ ∈ Ball ( M ( A ) sa ) in a Borel fashion from u and v such that ( u ⊕ v ) ≡ e iy · · · e iy ℓ ( uv ⊕
1) mod M ( J ) ,where ℓ ≥ does not depend on A , J and u, v .Proof. Fix a unitary path ( W t ) t ∈ [0 , in U ( M ( C )) from 1 to (cid:20) (cid:21) .Fix ℓ ≥ s, t ∈ [0 , k W s − W t k ≤ /
2. Then u t := ( u ⊕ W ∗ t ( v ⊕ W t ( uv ⊕ ∗ is a path of mod M ( J ) unitaries from 1 to ( u ⊕ v ) ( uv ⊕ ∗ to 1. Then, as in the proof of Lemma 2.35, us-ing Lemma 2.33 one can recursively choose in a Borel fashion y , . . . , y ℓ ∈ Ball ( M ( A ) sa ) such that e iy k ≡ u ∗ k/ℓ u ( k +1) /ℓ mod M ( J ) for k ∈ { , , . . . , ℓ − } and hence ( u ⊕ v ) ( uv ⊕ ∗ ≡ e iy · · · e iy ℓ mod M ( J ). (cid:3) The Definable Arveson Extension Theorem.
In the rest of this section, we present definable versions ofsome fundamental results in operator algebras, to be used in the development of definable K-homology. Supposethat H is a separable Hilbert space. We regard B ( H ) as the multiplier algebra of the C*-algebra K ( H ) of compactoperators on H . The corresponding strict topology on Ball ( B ( H )) is the strong-* topology. Consistently, weconsider B ( H ) as a standard Borel space with respect to the induced standard Borel structure. If Z is a separableBanach space, we consider L ( Z, B ( H )) as a strict Banach space, where Ball ( L ( Z, B ( H ))) is endowed with thetopology of pointwise strong-* convergence. We denote by U ( H ) the unitary group of B ( H ), which is a Polishgroup when endowed with the strong-* topology.Suppose that A is a separable unital C*-algebra, and X ⊆ A is an operator system [Pau02, Chapter 2]. Let H be a separable Hilbert space. Arveson’s Extension Theorem asserts that every contractive completely positive (ccp)map φ : X → B ( H ) [BO08, Section 1.5] admits a contractive completely positive extension ˆ φ : A → B ( H ) [Pau02,Theorem 7.5]. We observe now that ˆ φ can be chosen in a Borel way from φ . Notice that the space CCP ( X, B ( H ))of contractive completely positive maps is closed (hence, compact) in Ball ( L ( X, B ( H ))) endowed with the topologyof pointwise weak* convergence. Lemma 2.39.
Suppose that A is a separable unital C*-algebra, and X ⊆ A is an operator system. Let H be aseparable Hilbert space. Then there exists a Borel function
CCP (
X, B ( H )) → CCP (
A, B ( H )) , φ ˆ φ such that ˆ φ is an extension of φ . Towards the proof of Lemma 2.39, we recall the following particular case of the selection theorem for relationswith compact sections [Kec95, Theorem 28.8].
Lemma 2.40.
Suppose that
X, Y are compact metrizable spaces, and A ⊆ X × Y is a Borel subset such that, forevery x ∈ X , the vertical section A x = { y ∈ Y : ( x, y ) ∈ A } is a closed nonempty set. Then there exists a Borel function f : X → Y such that ( x, f ( x )) ∈ A for every x ∈ X . Using this selection theorem, Lemma 2.39 follows immediately from the Arveson Extension Theorem.
Proof of Lemma 2.39.
We consider CCP (
X, B ( H )) as a compact metrizable space, endowed with the topology ofpointwise weak* convergence. Consider the Borel set A ⊆ CCP (
X, B ( H )) × CCP (
A, B ( H )) of pairs ( φ, ψ ) suchthat ψ | X = φ . Then by the Arveson Extension Theorem, the vertical sections of A are nonempty, and clearly closed.Thus, by Lemma 2.40 there exists a Borel function f : CCP ( X, B ( H )) → CCP (
A, B ( H )) such that f ( φ ) | X = φ for every φ ∈ CCP (
X, B ( H )). (cid:3) The Definable Stinespring Dilation Theorem.
Suppose that A is a separable unital C*-algebra, and H is a separable Hilbert space. Stinespring’s Dilation Theorem asserts that, for every contractive completely positivemap φ : A → B ( H ), there exists a linear map V φ : H → H with k V φ k = k φ k and a nondegenerate representation π φ of A on H such that φ ( a ) = V ∗ φ π φ ( a ) V φ for every a ∈ A . Notice that the set Rep ( A, H ) of nondegeneraterepresentations of A on H is a G δ subset of Ball ( L ( A, B ( H ))), whence Polish with the subspace topology, whereBall ( B ( H )) is endowed with the strong-* topology. It follows from the proof of the Stinespring Dilation Theorem,where V and π are explicitly defined in terms of φ , that they can be chosen in a Borel way from φ ; see [Bla06,Theorem II.6.9.7] Lemma 2.41.
Suppose that A is a separable unital C*-algebra, and H is a separable Hilbert space. Then there existsa Borel function CCP (
A, B ( H )) → Ball ( B ( H )) × Rep (
A, B ( H )) , φ ( V φ , π φ ) such that φ ( a ) = V ∗ φ π φ ( a ) V φ for a ∈ A and k V φ k = k φ k for every contractive completely positive map φ : A → B ( H ) . The Definable Voiculescu Theorem.
Suppose that A is a separable unital C*-algebra, and ρ, ρ ′ : A → B ( H ) are two maps. If U ∈ U ( H ) is a unitary operator, write ρ ′ ≈ U ρ if ρ ′ ( a ) ≡ U ∗ ρ ( a ) U mod K ( H ) forevery a ∈ A . If V : H → H is an isometry, write ρ ′ . V ρ if ρ ′ ( a ) ≡ V ∗ ρ ( a ) V mod K ( H ) for every a ∈ A . Anondegenerate representation ρ of A on B ( H ) is ample if, for every a ∈ B ( H ), ρ ( a ) ∈ K ( H ) ⇒ a = 0. Noticethat the set ARep( A, H ) of ample representations of A on B ( H ) is a G δ subset of Ball( L ( A, B ( H ))). Similarly,the set Iso ( H ) of isometries H → H is a G δ subset of Ball ( B ( H )). A formulation of Voiculescu’s Theorem assertsthat if ρ : A → B ( H ) is an ample representation, and σ : A → B ( H ) is a unital completely positive (ucp) map, EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 21 then there exists an isometry V : H → H such that σ . V ρ ; see [HR00, Theorem 3.4.3, Theorem 3.4.6, Theorem3.4.7]. We will observe that one can select V in a Borel fashion from ρ and σ . Lemma 2.42.
Let A be a separable unital C*-algebra, and H a separable Hilbert space. There exists a Borelfunction UCP (
A, B ( H )) × ARep (
A, H ) → Iso ( H ) , ( σ, ρ ) V σ,ρ such that σ . V σ,ρ ρ . Towards obtaining a proof of Lemma 2.42, we argue as in the proof of Voiculescu’s theorem as expounded in[HR00, Chapter 3]. First, one considers the case of ucp maps A → B ( H ) where H is finite-dimensional. Thefollowing can be seen a definable version of [HR00, Proposition 3.6.7]. Notice that the set Proj ( H ) of orthogonalprojections H → H is closed subset of Ball ( B ( H )). Let Proj fd ( H ) be the Borel subset of finite-dimensionalprojections. The following lemma is a consequence of [HR00, Proposition 3.6.7] itself and the Luzin–NovikovUniformization Theorem for Borel relations with countable sections [Kec95, Theorem 18.10]. Lemma 2.43.
Fix a finite-dimensional subspace H of H , and regard B ( H ) as a C*-subalgebra of B ( H ) . For everyfinite subset F of A and ε > , there exists a Borel map UCP (
A, B ( H )) × ARep (
A, H ) × Proj fd ( H ) → Ball ( H ) , ( σ, ρ, P ) V such that Ran( V ) is orthogonal to P ( H ) and k σ ( a ) − V ∗ ρ ( a ) V k < ε for a ∈ F . One then uses Lemma 2.43 to establish Lemma 2.42 in the case of block-diagonal maps. Recall that σ is block-diagonal with respect to ( P n ) n ∈ ω if ( P n ) n ∈ ω is a sequence of pairwise orthogonal finite-rank projections P n ∈ B ( H )such that P n P n = I and σ ( a ) = P n P n σ ( a ) P n for every a ∈ A (where the convergence is in the strong-*topology). Consider the set BlockUCP ( A, B ( H )) of pairs (cid:0) σ, ( P n ) n ∈ ω (cid:1) ∈ UCP (
A, B ( H )) × Proj fd ( H ) ω such that σ is block-diagonal with respect to ( P n ) n ∈ ω . The proof of [HR00, Lemma 3.5.2] shows the following. Lemma 2.44.
There exists a Borel function
BlockUCP (
A, B ( H )) × ARep (
A, H ) → Iso ( H ) , (cid:0) σ, ( P n ) n ∈ ω , ρ (cid:1) V such that σ . V ρ . Finally, one shows that the general case of Voiculescu’s theorem can be reduced to the block-diagonal case, as in[HR00, Theorem 3.5.5].
Lemma 2.45.
There exists a Borel function
UCP (
A, B ( H )) → BlockUCP (
A, B ( H )) × Iso ( H ) , σ (cid:0) σ ′ , ( P n ) n ∈ ω , V ′ (cid:1) such that σ . V ′ σ ′ . Lemma 2.42 is then obtained by combining Lemma 2.44 and Lemma 2.45.As a consequence of the definable Voiculescu Theorem, one obtains the following; see [HR00, Theorem 3.4.6].
Lemma 2.46.
Let A be a separable unital C*-algebra, and H a separable Hilbert space. There exist: • a Borel map Rep (
A, H ) × ARep (
A, H ) → U ( H ) , ( ρ ′ , ρ ) U ρ ′ ,ρ such that ρ ′ ⊕ ρ ≈ U ρ ′ ,ρ ρ ; • a Borel map ARep (
A, H ) × ARep (
A, H ) → U ( H ) , ( ρ ′ , ρ ) W ρ ′ ,ρ such that ρ ≈ W ρ ′ ,ρ ρ ′ . Spectrum.
Suppose now that A is a strict unital C*-algebra, and J is a norm-separable closed two-sided idealof A . One can consider the quotient C*-algebra A /J and, for a ∈ A , the spectrum σ A /J ( a ) of a + J in A /J . Wealso let the resolvent ρ A /J ( a ) be the complement in C of σ A /J ( a ). The following lemma is analogous to [AM15,Theorem 3.16]. Lemma 2.47.
Suppose that A is a strict C*-algebra, and J a norm-separable closed two-sided ideal of A . Supposethat every invertible self-adjoint element of A /J lifts to an invertible self-adjoint element of A . If a ∈ A sa , and J is a countable dense subset of J ∩ A sa , then σ A /J ( a ) = \ d ∈ J σ ( a + d ) .Proof. It suffices to prove that ρ A /J ( a ) is the union of ρ ( a + d ) for d ∈ J . Clearly, ρ ( a + d ) ⊆ ρ A /J ( a ) for every d ∈ J , so it suffices to prove the other inclusion. Suppose that λ ∈ ρ A /J ( a ) ∩ R . We want to show that λ ∈ ρ ( a + d )for some d ∈ J . After replacing a with a − λ , it suffices to consider the case when λ = 0. In this case, a + J isinvertible in A /J . Therefore, by assumption there exists d ∈ J ∩ A sa such that a + d is invertible in A . Since theset of invertible elements of A is norm-open, there exists d ∈ J such that a + d is invertible in A , and hence0 ∈ ρ ( a + d ). (cid:3) Lemma 2.48.
Suppose that A is a strict unital C*-algebra, and J a norm-separable closed two-sided ideal of A .Suppose that every invertible self-adjoint element of A /J lifts to an invertible self-adjoint element of A . Then thefunction A sa → Closed ( R ) , a σ A /J ( a ) is Borel.Proof. Fix a countable norm-dense subset J of J ∩ A sa . Then by the previous lemma we have that, for a ∈ A sa , σ A /J ( a ) = \ d ∈ J σ ( a + d ) .As the function Closed ( R ) ω → Closed ( R ), ( F n ) T n ∈ ω F n is Borel, this concludes the proof. (cid:3) Lemma 2.49.
The function B ( H ) sa → K ( R ) , T σ ess ( T ) = σ Q ( H ) ( T ) is Borel.Proof. An operator T ∈ B ( H ) induces an invertible element of Q ( H ) if and only if it is Fredholm. If T is Fredholmand self-adjoint, then it has index 0, and 0 is an isolated point of the spectrum of T that is an eigenvalue withfinite multiplicity. Thus, if P is the finite-rank projection onto the eigenspace of 0 for T , then we have that T + P is invertible and self-adjoint and induces the same element of Q ( H ) as T . This shows that every invertible self-adjoint element of Q ( H ) lifts to an invertible self-adjoint element of B ( H ). Therefore, the conclusion follows fromProposition 2.48. (cid:3) Suppose that T ∈ Ball ( B ( H )) is a mod K ( H ) projective. Recall that this means that T is a positive operatorsatisfying T ≡ T mod K ( H ). Then it is well-known that there exists a projection P ≡ T mod K ( H ). We observethat one can choose such a P in a Borel fashion from T ; see [And20, Lemma 3.1]. Lemma 2.50.
Consider the Borel set
Proj ( B ( H ) /K ( H )) of mod K ( H ) projections in B ( H ) . Then there isa Borel function Proj ( B ( H ) /K ( H )) → Proj ( B ( H )) , T P T such that T ≡ P T mod K ( H ) for every T ∈ Proj ( B ( H ) /K ( H )) .Proof. Suppose that T ∈ Proj ( B ( H ) /K ( H )). Observe σ ess ( T ) ⊆ { , } . In particular, σ ess ( T ) is countable,with only accumulation points 0 and 1. From Lemma 2.49, the maps T σ ess ( T ) and T σ ( T ) are Borel. If σ ess ( T ) = { } then one can set P T = 0. If σ ess ( T ) = { } , one can set P T = 1.Let us consider the case when { , } = σ ess ( T ). By [Kec95, Theorem 12.13] there exists a Borel map Proj ( B ( H ) /K ( H )) → [0 , ω , T ( t n ) such that ( t n ) is an increasing enumeration of σ ( T ). One can then choose in a Borel way n ∈ ω such that t n < t n +1 and then a continuous function f : [0 , → [0 ,
1] such that f ( t i ) = (cid:26) i ≤ n ,1 if i ≥ n + 1.One can then set P T = f ( T ). (cid:3) Polar decompositions.
We now observe that the polar decomposition of an operator is given by a Borelfunction. We will use the following version of the selection theorem for relations with compact sections from [Kec95,Theorem 28.8].
Lemma 2.51.
Suppose that X is a standard Borel space, Y is a compact metrizable space, and A ⊆ X × Y is aBorel subset such that, for every x ∈ X , the vertical section A x = { y ∈ Y : ( x, y ) ∈ A } is a closed nonempty set. Then the assignment X → Closed ( Y ) , x A x , is Borel, where Closed ( Y ) is endowedwith the Effros Borel structure. As an application, we obtain the following. Let H be a separable Hilbert space. We consider the unit ballBall ( H ) of H as a compact metrizable space endowed with the weak topology. We also consider Closed (Ball ( H ))as a standard Borel space, endowed with the Effros Borel structure. Lemma 2.52.
The function B ( H ) → Closed (Ball ( H )) , T Ker ( T ) ∩ Ball ( H ) , is Borel.Proof. By Lemma 2.51, it suffices to show that the set A = { ( T, x ) ∈ B ( H ) × Ball ( H ) : T x = 0 } EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 23 is Borel. Fix a countable norm-dense subset { x n : n ∈ ω } of Ball ( H ). Then we have that, if ( T, x ) ∈ Ball ( B ( H )) × Ball ( H ), then ( T, x ) ∈ A if and only if ∀ k ∈ ω ∃ n ∈ ω such that k x − x n k < − k and k T x k k < − k . Since the normon Ball ( H ) is weakly lower-semicontinuous, this shows that A is Borel. (cid:3) Recall that, for an operator T ∈ B ( H ), one sets | T | := ( T ∗ T ) / . By strong-* continuity on bounded sets ofcontinuous functional calculus, the function T
7→ | T | is Borel. Furthermore, there exists a unique partial isometry U with Ker ( U ) = Ker ( T ) such that T = U | T | [Ped89, Theorem 3.2.17]. The decomposition T = U | T | is thencalled the polar decomposition of T . Lemma 2.53.
The function B ( H ) → B ( H ) , T U that assigns to an operator the partial isometry U in thepolar decomposition of T is Borel.Proof. It suffices to notice that is graph, which is the set of pairs (
T, U ) such that U is a partial isometry withKer ( U ) = Ker ( T ) and T = U | T | , is Borel by Lemma 2.52. (cid:3) Consider the Borel set U ( B ( H ) /K ( H )) of mod K ( H ) unitaries in B ( H ). Thus, T ∈ U ( B ( H ) /K ( H )) ifand only if T ∗ T ≡ T T ∗ ≡ I mod K ( H ). If U is the partial isometry in the polar decomposition of T , then U ≡ T mod K ( H ) and U is an essential unitary. In fact, one can easily define (in a Borel fashion from T ) anisometry or co-isometry V such that T ≡ V mod K ( H ). One has that T is in particular a Fredholm operator. Itsindex is defined by index ( T ) = rank (1 − V ∗ V ) − rank (1 − V V ∗ ) .Thus, index ( T ) is a Borel function of T ∈ U ( B ( H ) /K ( H )).More generally, consider the Borel set of pairs ( P, T ) ∈ Ball ( B ( H )) such that P is a projection, P T = T P = T and T T ∗ ≡ T ∗ T ≡ P . If V is the partial isometry in the polar decomposition of T , then V ≡ T mod K ( H ) andthe index of P T P regarded as a Fredholm operator on
P H is given by the Borel functionindex (
P T P ) = rank ( P − V ∗ V ) − rank ( P − V V ∗ ) .3. K -theory of unital C*-algebras with a strict cover In this section we explain how the K and K groups of a unital C*-algebra with a strict cover can be regardedas semidefinable groups. We also recall the definition of the index map and the exponential map between theK and K groups, and observe that they are definable homomorphisms. Finally, we consider the six-term exactsequence associated with a strict unital C*-pair, and observe that the connective maps are all definable grouphomomorphisms.3.1. K -group. Suppose that A / J is a unital C*-algebra with a strict cover. Recall that Proj ( A / J ) denotes thePolish space of mod J projections in A . Similarly, for n ≥ M n ( A / J )) := Proj ( M n ( A ) /M n ( J ))is a Polish space. We say that an element of Proj ( M n ( A / J )) for some n ≥ J projection over A . Wedefine Z ( A / J ) be the set of pairs of mod J projections over A , which is the disjoint union of Z ( n )0 ( A / J ) :=Proj ( M n ( A / J )) × Proj ( M n ( A / J )) for n ≥ J projections p, q in A are Murray–von Neumann equivalent (respectively, unitary equivalent, and homotopic) mod J if and only if p + J and q + J are Murray–von Neumann equivalent (respectively, unitary equivalent, and homotopic)in A / J .The K -group K ( A / J ) of A / J —see [HR00, Chapter 4]—is defined as a quotient of Z ( A / J ) by an equivalencerelation B ( A / J ), defined as follows. For ( p, p ′ ) , ( q, q ′ ) ∈ Z ( A / J ), ( p, p ′ ) B ( A / J ) ( q, q ′ ) if and only if there exist m, n ∈ ω and r ∈ Proj ( M m ( A / J )) such that p ⊕ q ′ ⊕ r ⊕ n and q ⊕ p ′ ⊕ r ⊕ n are Murray–von Neumann equivalentmod J . By Lemma 2.37, we have the following equivalent description of B ( A / J ). Lemma 3.1.
Suppose that A / J is a unital C*-algebra with a strict cover, and ( p, p ′ ) , ( q, q ′ ) ∈ Z ( A / J ) where p, p ′ ∈ M d ( A / J ) and q, q ′ ∈ M k ( A / J ) . Then ( p, p ′ ) B ( A / J ) ( q, q ′ ) if and only if there exist m, n ∈ ω and y , . . . , y ℓ ∈ Ball ( M d + k + m + n ( A ) sa ) such that, setting u := e iy · · · e iy ℓ , one has that u ( p ⊕ q ′ ⊕ m ⊕ n ) u ∗ ≡ q ⊕ p ′ ⊕ m ⊕ n mod J ,where ℓ ≥ does not depend on A / J and ( p, p ′ ) , ( q, q ′ ) ∈ Z ( A / J ) . The (commutative) group operation on K ( A / J ) is induced by the Borel function on Z ( A / J ), (( p, p ′ ) , ( q, q ′ )) ( p ⊕ q, p ′ ⊕ q ′ ). The neutral element of K ( A / J ) corresponds to (0 , ∈ Z ( A / J ). The function that maps anelement to its additive inverse is induced by the Borel function on Z ( A / J ) given by ( p, p ′ ) ( p ′ , p ). Thus,K ( A / J ) is in fact a semidefinable group.If A / I and B / J are unital C*-algebras with a strict cover, and ϕ : A / I → B / J is a definable unital *-homomorphism, then the induced group homomorphism K ( A / I ) → K ( B / J ) is also definable. Thus, the as-signment A / J → K ( A / J ) gives a functor from the category of unital C*-algebras with a strict cover to thecategory of semidefinable abelian groups.Suppose that ( A , J ) is a strict unital C*-pair. We denote by J + the unitization of J , which can be identified withthe C*-subalgebra J + = span { J , } ⊆ A . Since J is a proper ideal of A , we can write every element of J + uniquelyas a + λ a ∈ J and λ ∈ C . More generally, every element of M n ( J + ) can be written uniquely as a + α a ∈ M n ( J ) and α ∈ M n ( C ). As the map M n ( J + ) → M n ( C ), a + α α is a unital *-homomorphism, wehave that k α k ≤ k a + α k and hence k a k ≤ k a + α k for a + α ∈ M n ( J + ).We define Proj( M n ( J + )) to be the set of projections in M n ( J + ), which we regard as a Borel subset of 2Ball ( M n ( J )) × Ball ( M n ( C )). Similarly, the unitary group U ( M n ( J + )) of M n ( J + ) is regarded as a Borel subset of 2Ball ( M n ( J )) × Ball ( M n ( C )). Define also Z ( n )0 ( J ) to be the Borel subset of Proj( M n ( J + )) × Proj( M n ( J + )) consisting of pairs ( p, p ′ )such that p ≡ p ′ mod M n ( J ). Finally, let Z ( J ) to be the disjoint union of Z ( n )0 ( J ) for n ≥ -group K ( J ) of J —see [HR00, Definition 4.2.1]—is defined as a quotient of Z ( J ) by an equivalencerelation B ( J ), defined as follows. One has that, for ( p, p ′ ) , ( q, q ′ ) ∈ Z ( J ), ( p, p ′ ) B ( J ) ( q, q ′ ) if and only if thereexist m, n ∈ ω and x ∈ Proj( M m ( J + )) such that p ⊕ q ′ ⊕ x ⊕ n and q ⊕ p ′ ⊕ x ⊕ n ′ are Murray–von Neumannequivalent. For ( p, p ′ ) ∈ Z ( J ), we let [ p ] − [ p ′ ] be the corresponding element of K ( J ). The (commutative) groupoperation on K ( J ) is induced by the Borel function on Z ( J ), (( p, p ′ ) , ( q, q ′ )) ( p ⊕ q, p ′ ⊕ q ′ ). The neutralelement of K ( J ) corresponds to (0 , ∈ Z ( J ). The function that maps an element of K ( J ) to its additive inverseis induced by the Borel function on Z ( J ) given by ( p, p ′ ) ( p ′ , p ). Thus, K ( J ) is a semidefinable group.If ( A , I ) are ( B , J ) are strict C*-pairs, and ϕ : ( A , I ) → ( B , J ) is a strict *-homomorphism, then it induces astrict *-homomorphism ϕ | I : I → J . In turn, this induces a definable group homomorphism K ( J ) → K ( I ). Thisgives a functor ( A , I ) K ( I ) from strict unital C*-pairs to semidefinable groups. Lemma 3.2.
Suppose that ( A , J ) is a strict unital C*-pair. Then there is a Borel map Z ( J ) → Z ( J ) , ( P, P ′ ) ( p, p ′ ) such that [ P ] − [ P ′ ] = [ p ] − [ p ′ ] and p ′ ∈ M n ( C ) .Proof. Suppose that (
P, P ′ ) ∈ Z ( d )0 ( J ). By definition, we have that for some x, x ′ ∈ M d ( J ) and α ∈ M d ( C ), P = x + α and P ′ = x ′ + α . Thus, we can define p := (cid:20) P
00 1 d − P ′ (cid:21) ∈ M d (cid:0) J + (cid:1) and p ′ := (cid:20) α
00 1 d − α (cid:21) ∈ M d ( C ) .Then we have that [ p ] − [ p ′ ] = [ P ] + [1 d − P ′ ] + [ α ] − [1 d − α ] = [ P ] − [ P ′ ] .This concludes the proof. (cid:3) Relative K -group. Suppose now that ( A , J ) is a strict unital C*-pair. For n ≥
1, define Z ( n )0 ( A , A / J ) tobe the Borel set of triples ( p, q, x ) ∈ Ball ( A ) where p, q are projections and x ∈ Ball ( M n ( A )) satisfies x ∗ x ≡ p mod M n ( J ) and xx ∗ ≡ q mod M n ( J ). Define Z ( A , A / J ) to be the disjoint union of K ( n )0 ( A , A / J ) for n ≥ ( A , A / J ) are called relative K-cycles for( A , A / J ); see [HR00, Definition 4.3.1]. If ( p, q, x ) ∈ Z ( n )0 ( A , A / J ), then we say that ( p, q, x ) is a relative K-cycle ofdimension n . A relative K-cycle ( p, q, x ) for ( A , A / J ) is degenerate if x ∗ x = p and xx ∗ = q . Two relative K-cycles( p, q, x ) and ( p ′ , q ′ , x ′ ) of dimension n are homotopic if there exists a norm-continuous path (( p t , q t , x t )) t ∈ [0 , ofrelative K-cycles for ( A , A / J ) of dimension n with ( p, q, x ) = ( p , q , x ) and ( p ′ , q ′ , x ′ ) = ( p , q , x ). EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 25
Notice that if ( p, q, x ) is a relative K-cycle of dimension d , and ( u t ) t ∈ [0 , is a path of unitaries in M d ( A ) startingat 1, then ( u ∗ t pu t , u ∗ t qu t , u ∗ t xu t )and ( p, u ∗ t qu t , u ∗ t x )are norm-continuous paths of relative K-cycles starting at ( p, q, x ). If p ≡ q ≡ x mod M d ( J ), then( p, q, tp + (1 − t ) q )is a norm-continuous path of relative cycles from ( p, q, x ) to ( p, q, p ). We have the following lemma; see [NdK17,Proposition 3.4]. Lemma 3.3.
Suppose that ( p, q, x ) is a relative cycle of dimension n for ( A , A / J ) . Then r := ( p ⊕ q, q ⊕ p, x ⊕ x ∗ ) is homotopic to the degenerate cycle ( p ⊕ q, p ⊕ q, p ⊕ q ) . The relative K -group K ( A , A / J ) is defined to be the quotient of Z ( A , A / J ) by the equivalence relationB ( A , A / J ) defined as follows. For ( p, q, x ) , ( p ′ , q ′ , x ′ ), set ( p, q, x ) B ( A , A / J ) ( p ′ , q ′ , x ′ ) if and only if there exists a degenerate relative K-cycles ( p , q , x ) , ( p ′ , q ′ , x ′ ) such that ( p ⊕ p , q ⊕ q , x ⊕ x ) and ( p ′ ⊕ p ′ , q ′ ⊕ q ′ , x ′ ⊕ x ′ )are of the same dimension and homotopic. The group operations on K ( A , A / J ) are induced by the Borel maps(( p, q, x ) , ( p ′ , q ′ , x ′ )) ( p ⊕ p ′ , q ⊕ q ′ , x ⊕ x ′ )and ( p, q, x ) ( q, p, x ∗ ) .It follows from Lemma 3.3 that K ( A , A / J ) is indeed a group. We let [ p, q, x ] be the element of K ( A , A / J )represented by the relative K-cycle ( p, q, x ). The trivial element of K ( A , A / J ) is equal to [ p, q, x ] where ( p, q, x ) isany degenerate relative K-cycle. Let J + be the unitization of J , which we identify with span ( J , ⊆ A . Lemma 3.4.
There is a Borel function Z ( A , A / J ) → Z ( A , A / J ) , ( P, Q, X ) ( p, q, p ) such that p ∈ M n ( C ) , q ∈ M n ( J + ) , p ≡ q ∈ M n ( J ) , and [ P, Q, X ] = [ p, q, p ] ∈ K ( A , A / J ) .Proof. Notice that (1 − P, − P, − P ) is a degenerate relative K-cycle of dimension d . Consider then( P ⊕ (1 − P ) , Q ⊕ (1 − P ) , X ⊕ (1 − P )) . By Lemma 2.38 one can choose Y , . . . , Y ℓ ∈ Ball ( M d ( A ) sa ) in a Borel way such that, setting U := e iY · · · e iY ℓ , onehas that U ( P ⊕ (1 − P ) U ∗ ) = 1 d ⊕ d , where ℓ ≥ A , J and ( P, Q, X ). Thus, after replacing(
P, Q, X ) with ( U ( P ⊕ (1 − P )) U ∗ , U ( Q ⊕ (1 − P )) U ∗ , U ( X ⊕ (1 − P )) U ∗ ) , we can assume without loss of generality that P = 1 d ⊕ d .By Lemma 2.35 one can choose Y , . . . , Y ℓ ∈ Ball ( M d ( A ) sa ) in a Borel fashion from ( P, Q, X ) such that, setting U := e iY · · · e iY ℓ , one has that U ∗ ( Q ⊕ d ) U ≡ ( P ⊕ d ) mod M d ( J ) .Thus, after replacing ( P, Q, X ) with ( P ⊕ d , U ∗ ( Q ⊕ d ) U, U ∗ ( X ⊕ d )) we can assume without loss of gener-ality that P = 1 d ⊕ d ∈ M d ( C ) and Q ∈ M d ( A ) satisfy P ≡ Q mod M d ( J ) and hence Q ∈ M n ( J + ).In this case, we have that [ P, Q, X ] = [
P, Q, P ], since ( P t , Q t , tP + (1 − t ) X ) t ∈ [0 , is a norm-continuous path ofrelative K-cycles from ( P, Q, X ) to (
P, Q, P ). This concludes the proof. (cid:3)
Proposition 3.5.
Suppose that ( A , J ) is a strict C*-pair. Then K ( A , A / J ) is a definable group. The assignment K ( J ) → K ( A , A / J ) , [ P ] − [ Q ] [ P, Q, P ] is a natural definable isomorphism, called the excision isomorphism .Proof. By [NdK17, Theorem 3.9], the excision homomorphism K ( J ) → K ( A , A / J ) is bijective; see also [HR00,Theorem 4.3.8]. Clearly, it is induced by a Borel function Z ( J ) → Z ( A , A / J ). By Lemma 3.4 the inversehomomorphism K ( A , A / J ) → K ( J ) is also induced by a Borel function Z ( A , A / J ) → Z ( J ). Thus, K ( A , A / J )is a definable group, and the excision isomorphism is a definable isomorphism. (cid:3) There is a natural definable homomorphism K ( A , A / J ) → K ( A ) that is induced by the Borel map Z ( A , A / J ) → Z ( A ) ( p, q, x ) ( p, q ). We also have a natural definable homomorphism K ( A ) → K ( A / J ) induced by the Borelmap Z ( A ) → Z ( A / J ). We have the following result; see [HR00, Proposition 4.3.5]. Proposition 3.6.
Suppose that ( A , J ) is a strict unital C*-pair. The (natural) sequence of definable groups anddefinable group homomorphisms K ( A , A / J ) → K ( A ) → K ( A / J ) is exact. Combining the excision isomorphism K ( J ) → K ( A , A / J ) with the natural definable homomorphism K ( A , A / J ) → K ( A ), we obtain a natural definable group homomorphism K ( J ) → K ( A ). This is defined by mapping( p, q ) ∈ Z ( n )0 ( J ) to ( p, q ) regarded as an element of Z ( n )0 ( A ). Combining Proposition 3.5 with Proposition 3.6we have the following. Corollary 3.7.
Suppose that A is a unital strict C*-algebra and J is a proper strict ideal of A . Then the naturalsequence K ( J ) → K ( A ) → K ( A / J ) is exact. group. Suppose that ( A , J ) is a strict unital C*-pair. We can then consider the Borel set U ( A / J ) ofelements of Ball ( A ) that are unitaries mod J . We then let Z ( A / J ) to be the disjoint union of U ( M n ( A ) /M n ( J )) for n ≥
1. The equivalence relation B ( A / J ) on Z ( A / J ) is defined by setting u B ( A / J ) u ′ for u ∈ U ( M n ( A ) /M n ( J ))and u ′ ∈ U ( M n ′ ( A ) /M n ′ ( J )) if and only if there exist k, k ′ ∈ ω with n + k = n ′ + k ′ and such that there is a norm-continuous path from u ⊕ k + M n + k ( J ) to u ′ ⊕ k ′ + M n + k ( J ) in the unitary group of the quotient unitalC*-algebra M n + k ( A ) /M n + k ( J ). This equivalence relation is analytic by Corollary 2.34. The definable K -groupK ( A / J ) is then the semidefinable group obtained as quotient Z ( A / J ) / B ( A / J ) with group operations defined asabove. This defines a functor A / J K ( A / J ) from unital C*-algebras with strict cover to semidefinable groups.Given a strict unital C*-pair ( A , J ), we also consider the definable K -group K ( J ) of J . As above, we identifythe unitization J + of J with span { J , } ⊆ A . For n ≥ U ( M d ( J + )) be the unitary group of M d ( J + ). Recallthat every element of M d ( J + ) can be written uniquely as x + α x ∈ M d ( J ) and α ∈ M d ( C ). We consider U ( M d ( J + )) as a Borel subset of Ball ( M d ( J )) × Ball ( M d (( C ))). We then set Z ( J ) to be the disjoint union of U ( M d ( J + )) for d ≥
1, and let B ( J ) be the (analytic) equivalence relation on Z ( J ) obtained by setting u B ( J ) u ′ for u ∈ U ( M n ( J + )) and u ′ ∈ U ( M n ′ ( J + )) if and only if there exist k, k ′ ∈ ω with n + k = n ′ + k ′ and such thatthere is a norm-continuous path from u ⊕ k to u ′ ⊕ k ′ in U ( M n + k ( J + )). The definable K -group K ( J ) is thenthe semidefinable group obtained as quotient Z ( J ) / B ( J ) with group operations defined as above.Suppose that ( A , J ) is a strict unital C*-pair. We have natural definable group homomorphismsK ( J ) → K ( A ) → K ( A / J ) .The definable group homomorphism K ( J ) → K ( A ) is induced by the inclusion J + ⊆ A , which gives an inclusion U ( M d ( J + )) → U ( M d ( A )) for every d ≥
1. The definable group homomorphism K ( A ) → K ( A / J ) is also inducedby the inclusion maps U ( M d ( A )) → U ( M d ( A ) /M d ( J )) for d ≥
1. We have the following result, which can beeasily verified directly, and also follows from Corollary 3.7 via the Bott isomorphism theorem [HR00, Theorem4.9.1].
Proposition 3.8.
Suppose that ( A , J ) is a strict unital C*-pair. The sequence of natural definable homomorphisms K ( J ) → K ( A ) → K ( A / J ) is exact. The six-term exact sequence.
Suppose that ( A , J ) is a strict unital C*-pair. One can define a naturaldefinable group homomorphism ∂ : K ( A / J ) → K ( J ) called the index map , as follows. An element of K ( A / J )is of the form [ u ] where u ∈ U ( M d ( A ) /M d ( J )) for some d ≥
1. Then define P := " uu ∗ u (1 − u ∗ u ) / u ∗ (1 − uu ∗ ) / − u ∗ u ∈ M d (cid:0) J + (cid:1) EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 27 and Q := 1 d ⊕ d ∈ M d (cid:0) J + (cid:1) .Then P, Q are projections such that P ≡ Q mod M d ( J ) and hence ( P, Q ) ∈ Z ( J ). One then defines ∂ ([ u ]) =[ P ] − [ Q ]; see [HR00, Proposition 4.8.10]. As ( P, Q ) is obtained in a Borel fashion from u , the boundary map ∂ : K ( A / J ) → K ( J ) is definable.Equivalently, one can define ∂ as follows. Given an element [ u ] of K ( A / J ) for some u ∈ U ( M d ( A ) /M d ( J )).Consider the partial isometry v ∈ M d ( A ) defined by v = (cid:20) u − u ∗ u ) / (cid:21) and observe that v ≡ u ⊕ J [RLL00, Lemma 9.2.1]. Then 1 d − v ∗ v and 1 d − vv ∗ are projections in M d ( J + )such that 1 d − v ∗ v ≡ d − vv ∗ ≡ d ⊕ d ∈ M d ( C ). Therefore, (1 d − v ∗ v, d − vv ∗ ) ∈ Z ( J ). One has that ∂ [ u ] = [1 d − v ∗ v ] − [1 d − vv ∗ ] ∈ K ( J ); see [RLL00, Proposition 9.2.3]. Then we have the following; see [RLL00,Lemma 9.3.1 and Lemma 9.3.2]. Proposition 3.9.
Suppose that A is a strict unital C*-algebra and J is a strict ideal of A . Then the sequence K ( A ) → K ( A / J ) ∂ → K ( J ) → K ( A ) is exact. Suppose that ( A , J ) is a strict unital C*-pair. One can consider a natural definable homomorphism ∂ :K ( A / J ) → K ( J ) called the exponential map . This is defined as follows. Consider an element of K ( A / J ) ofthe form [ p ] − [ q ] for some p, q ∈ Proj ( M n ( A ) /M n ( J )). Then we have that exp (2 πip ) and exp (2 πiq ) are uni-tary elements of M n ( J + ) such that exp (2 πip ) ≡ exp (2 πiq ) mod M n ( J ). Then one has that ∂ ([ p ] − [ q ]) =[exp (2 πip )] − [exp (2 πiq )] ∈ K ( J ); see [RLL00, Proposition 12.2.2] and [HR00, Section 4.9]. From Proposition 3.9one can obtain via the Bott isomorphism theorem [HR00, Theorem 4.9.1] the following. Proposition 3.10.
Suppose that ( A , J ) is a strict unital C*-pair. Then the sequence K ( A ) → K ( A / J ) ∂ → K ( J ) → K ( A ) is exact. Suppose that ( A , J ) is a strict unital C*-pair. Then as discussed above we have exact sequencesK ( J ) → K ( A ) → K ( A / J )and K ( J ) → K ( A ) → K ( A / J ) .These are joined together by the index and exponential maps. From Proposition 3.10, Proposition 3.9, Corollary3.7 and Proposition 3.8, one obtains the six-term exact sequenceK ( J ) K ( A ) K ( A / J )K ( A / J ) K ( A ) K ( J ) ∂ ∂ for the strict unital C*-pair ( A , J ), where the vertical arrows are the index map and the exponential map; see[RLL00, Theorem 12.1.2]. 4. Definable K -homology of separable C*-algebras In this section we recall the definition of the Ext invariant for separable unital C*-algebras, and its descriptiondue to Paschke in terms of the K-theory of Paschke dual algebras as defined in [Hig95, HR00] or, equivalently,of commutants in the Calkin algebra. Following [HR00, Chapter 3], we consider the group Ext( − ) − defined interms of unital semi-split extensions. In the case of separable unital nuclear C*-algebras, every unital extension issemi-split, and the group Ext( − ) − coincides with the group Ext( − ) defined in terms of unital extensions. Using Paschke’s K-theoretical description of Ext from [Pas81], we show that Ext ( − ) − yields a contravariant functor fromseparable unital C*-algebras to the category of definable groups.We also recall the definition of the K-homology groups of separable C*-algebras as in [HR00, Chapter 5]. Usingtheir description in terms of Ext, we conclude that they can be endowed with the structure of definable groups ,in such a way that the assignments A K ( A ) and A K ( A ) are functors from the category of separableC*-algebras to the category of definable groups.We define a separable C*-pair to be a pair ( A, I ) where A is a separable C*-algebra and I is a closed two-sidedideal of A . A morphism ( A, I ) → ( B, J ) between separable C*-pairs is a *-homomorphisms A → B that maps I to J . Recall that a C*-algebra A is nuclear if the identity map of A is the pointwise limit of contractive completelypositive maps that factor through finite-dimensional C*-algebras; see [HR00, Section 3.3]. We say that a separableC*-pair ( A, I ) is nuclear if A is nuclear. In this section, we will also introduce the relative definable K-homologygroups, and the six-term exact sequence in K-homology associated with a separable nuclear C*-pair.4.1. C*-algebra extensions and the Ext group.
Let H be a separable Hilbert space, and B ( H ) be the algebraof bounded linear operators on H . We let K ( H ) ⊆ B ( H ) be the closed ideal of compact operators, and Q ( H ) bethe Calkin algebra , which is the quotient of B ( H ) by K ( H ). Let π : B ( H ) → Q ( H ) be the quotient map.If U ∈ U ( H ) is a unitary operator, then U defines an automorphism Ad( U ) : B ( H ) → B ( H ) given by T U ∗ T U . As K ( H ) is Ad ( U )-invariant, we have an induced automorphism of Q ( H ), still denoted by Ad( U ).Suppose that A is a unital, separable C*-algebra. A unital extension of A (by K ( H )) is a unital *-homomorphism ϕ : A → Q ( H ). A unital extension of A is injective or essential if it is an injective *-homomorphism A → Q ( H ).Two extensions ϕ, ϕ ′ : A → Q ( H ) are equivalent if there exists U ∈ U ( H ) such that Ad( U ) ◦ ϕ ′ = ϕ . An injective,unital extension ϕ : A → Q ( H ) is semi-split (or weakly nuclear in the terminology of [EK01]) if there exists aunital completely positive (ucp) map σ : A → B ( H ) such that ϕ = π ◦ σ [HR00, Theorem 3.1.5]. An injectiveunital extension ϕ : A → Q ( H ) is split or trivial if there exists a unital *-homomorphism ˜ ϕ : A → B ( H ) such that ϕ = π ◦ ˜ ϕ .Every unital, essential extension ϕ : A → Q ( H ) determines an exact sequence0 → K ( H ) → E ϕ → A → E ϕ = { ( x, y ) ∈ A ⊕ B ( H ) : ϕ ( x ) = π ( y ) } .and K ( H ) is an essential ideal of E ϕ . The extension is split if and only if the map E ϕ → A is a split epimorphismin the category of unital C*-algebras and unital *-homomorphisms.Conversely, given an exact sequence 0 → K ( H ) → E p → A → p : E → A is a unital *-homomorphism and K ( H ) is an essential ideal of E , one can define an essential unitalextension ϕ : A → Q ( H ) as follows. Consider K ( H ) ⊆ E ⊆ B ( H ), then define ϕ ( a ) = π ( b a ) ∈ Q ( H ) for a ∈ A where b a ∈ E is such that p ( b a ) = a . Again, we have that ϕ is trivial if and only if p : E → A is a split epimorphism.Let A be a separable, unital C*-algebra. One defines Ext( A ) to be the set of unitary equivalence classes ofunital, injective extensions of A by K ( H ); see [HR00, Definition 2.7.1], and Ext nuc ( A ) = Ext ( A ) − to be thesubset of unitary equivalence classes of unital, injective semi-split (or weakly nuclear) extensions of A by K ( H )[Bla98, 15.7.2].One can define a commutative monoid operation on Ext ( A ). The (additively denoted) operation on Ext( A ) isinduced by the map ( ϕ, ϕ ′ ) Ad ( V ) ◦ ( ϕ ⊕ ϕ ′ ) where V : H → H ⊕ H is a surjective linear isometry; see [HR00,Proposition 2.7.2]. By Voiculescu’s Theorem [HR00, Theorem 3.4.3], one has that the neutral element of Ext ( A )is the set of split extensions, which form a single unitary equivalence class [HR00, Theorem 3.4.7]. Furthermore,the set Ext( A ) − is equal to the set of elements of Ext( A ) that have an additive inverse, whence it forms a group[HR00, Definition 2.7.6]. When A is a nuclear unital separable C*-algebra, by the Choi–Effros lifting theorem[HR00, Theorem 3.3.6], one has that every extension of A is semi-split, and Ext( A ) = Ext ( A ) − . In particular, inthis case Ext( A ) is itself a group.Let A be a separable unital C*-algebra. We regard Ext( A ) − as a definable group, as follows. Fix a separableHilbert space H . Let us say that a ucp map φ : A → B ( H ) is ample if k ( π ◦ φ ) ( x ) k = k x k for every x ∈ A .Notice that the set AUCP ( A, B ( H )) of ample ucp maps A → B ( H ) is a G δ subset of the space Ball ( L ( A, B ( H ))) EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 29 of bounded linear maps of norm at most 1 endowed with the topology of pointwise strong-* convergence. Thus,AUCP (
A, B ( H )) is a Polish space.Let E ( A ) ⊆ AUCP (
A, B ( H )) be the Borel set of ample ucp maps ϕ : A → B ( H ) such that ϕ ( xy ) ≡ ϕ ( x ) ϕ ( y ) mod K ( H )for x, y ∈ A . An injective, unital semi-split extension of A by definition has a ucp lift, which is an element of E ( A ), and conversely every element of E ( A ) gives rise to an injective, unital semi-split extension of A . Thus, wecan regard E ( A ) as the space of representatives of injective, unital semi-split extensions of A . We define a Polishtopology on E ( A ) that induces the Borel structure on E ( A ) by declaring a net ( ϕ i ) i ∈ I in E ( A ) to converge to ϕ ifand only if, for every x, y ∈ X , ( ϕ i ( x )) i ∈ ω strong-* converges to ϕ ( x ), and ( ϕ i ( xy ) − ϕ i ( x ) ϕ i ( y )) i ∈ ω norm-convergesto ϕ ( xy ) − ϕ ( x ) ϕ ( y ).Two elements ϕ, ϕ ′ of E ( A ) represent the same element [ ϕ ] of Ext( A ) − if and only if there exists U ∈ U ( H )such that U ∗ ϕ ( a ) U ≡ ϕ ′ ( a ) mod K ( H ) for every a ∈ A . This defines an analytic equivalence relation ≈ on E ( A ).We can thus regard Ext( A ) − as the semidefinable set E ( A ) / ≈ .We now observe that the group operations on Ext( A ) − are definable, and thus this turns Ext ( A ) − into asemidefinable group. We will later show in Proposition 4.12 that in fact Ext ( A ) − is a definable group. Proposition 4.1.
Let A be a separable unital C*-algebra. The addition operation ( x, y ) x + y and the additiveinverse operation x
7→ − x on Ext ( A ) − are definable functions. Thus, E ( A ) / ≈ = Ext ( A ) − is a semidefinablegroup.Proof. Fix a representation A ⊆ B ( H ) such that A ∩ K ( H ) = { } . The assertion for addition is clear, as the Borelmap E ( A ) × E ( A ) → E ( A ), ( ϕ, ϕ ′ ) Ad ( W ) ◦ ( ϕ ⊕ ϕ ′ ) is a lift for the addition operation, where W is a fixedsurjective linear isometry H → H ⊕ H .In order to obtain a lift for the function Ext ( A ) − → Ext ( A ) − , x
7→ − x , one can use the definable StinespringDilation Theorem (Lemma 2.41). Thus, if ϕ ∈ E ( A ), and π and V are the nondegenerate representation π of A and the isometry V : H → H obtained from ϕ in a Borel fashion as in Lemma 2.41, then defining the projection P := I − V V ∗ ∈ B ( H ) and ϕ ′ : A → B ( H ), a W ∗ ( P π ( a ) P ⊕ a ) W , one has that ϕ ′ ∈ E ( A ) represents − [ ϕ ],where as above W : H → H ⊕ H is a fixed surjective linear isometry; see also [HR00, Theorem 3.4.7]. (cid:3) Suppose that
A, B ⊆ B ( H ) are separable unital C*-algebras. A unital *-homomorphism α : A → B inducesa definable group homomorphism Ext ( B ) − → Ext ( A ) − , as follows. If ϕ ∈ E ( B ) is a representative for aninjective, unital extension, then one can consider α ∗ ( ϕ ) ∈ E ( A ) defined by a W ∗ (( ϕ ◦ α ) ( a ) ⊕ a ) W where W : H → H ⊕ H is a fixed surjective linear isometry. This defines a Borel function α ∗ : E ( B ) → E ( A ), whichinduces a definable group homomorphism α ∗ : Ext ( B ) − → Ext ( A ) − . Thus, Ext ( − ) − is a contravariant functorfrom the category of separable unital C*-algebras to the category of semidefinable groups.4.2. K -group and the Voiculescu property. Let A be a strict unital C*-algebra. Recall that two projections p, q ∈ A are Murray–von Neumann (MvN) equivalent if there exists v ∈ A such that v ∗ v = p and vv ∗ = q , in whichcase we write p ∼ MvN q . We say that a projection p ∈ A is ample if p ⊕ p ⊕
1, and co-ample if 1 − p is ample. Definition 4.2.
Let A be a strict unital C*-algebra. We say that A satisfies the Voiculescu property if the setof ample projections in A is a Borel subset of Ball ( A ) containing 1, and there exist strict unital *-isomorphismsΦ k,n : M n ( A ) → M k ( A ) for n, k ≥ n, k, m, n , k , n , k ≥ n > k , Φ k,n ( p ⊕ n − k ) ∼ MvN p for every projection p ∈ M k ( A );(2) Φ n,n = id M n ( A ) ,(3) Φ k,m ◦ Φ m,n is unitarily equivalent to Φ k,n ;(4) Φ k ,n ⊕ Φ k ,n is unitarily equivalent to Φ k + k ,n + n | M n ( A ) ⊕ M n ( A ) .We then define Z A0 ( A ) to be the Borel set of projections in A that are both ample and co-ample. Remark 4.3.
Let A be a strict C*-algebra that satisfies the Voiculescu property. Then for n > k ≥ p ∈ M n ( A )we have Φ k,n ( p ) ⊕ n − k ∼ MvN p . Indeed, by (1) we have that Φ k,n (Φ k,n ( p ) ⊕ n − k ) ∼ MvN Φ k,n ( p ). Therefore,Φ k,n ( p ) ⊕ n − k ∼ MvN p . Lemma 4.4.
Suppose that A is a strict unital C*-algebra that satisfies the Voiculescu property. Then [1] is theneutral element of K ( A ) .Proof. Recall that, by Lemma 3.1, given projections p, q over A , we have that [ p ] = [ q ] if and only if there exist m, n, n ′ ∈ ω such that p ⊕ m ⊕ n and q ⊕ m ⊕ n ′ are Murray–von Neumann equivalent. Suppose that p is aprojection over A . As 1 ⊕ ∼ MvN ⊕ p ⊕ ⊕ ∼ MvN ( p ⊕ ⊕ p and p ⊕ ( A ). Therefore, we have that[ p ] + [1] = [ p ⊕
1] = [ p ] .This shows that [1] is the neutral element of K ( A ). (cid:3) Lemma 4.5.
Suppose that A is a strict unital C*-algebra that satisfies the Voiculescu property. If p, q ∈ M n ( A ) ,then Φ , n +2 ( p ⊕ (1 − q ) ⊕ ⊕ ∈ Z A0 ( A ) and [ p ] − [ q ] = [Φ , n +2 ( p ⊕ (1 − q ) ⊕ ⊕ in K ( A ) .Proof. If p ∈ M n ( A ) and q ∈ M m ( A ) are projections over A , then we have that[ p ] − [ q ] = [ p ] + [1] − [ q ] = [ p ] + [1 − q ]= [ p ] + [1 − q ] + [1]= [ p ⊕ (1 − q ) ⊕ ⊕
0] .As, by Remark 4.3, Φ , n +2 ( p ⊕ (1 − q ) ⊕ ⊕ ⊕ n +1 ∼ MvN p ⊕ (1 − q ) ⊕ ⊕ p ] − [ q ] = [ p ⊕ (1 − q ) ⊕ ⊕
0] = [Φ , n +2 ( p ⊕ (1 − q ) ⊕ ⊕ , n +2 ( p ⊕ (1 − q ) ⊕ ⊕
0) is ample and co-ample. Set r := p ⊕ (1 − q ). We have by (4) ofDefinition 4.2 and since 1 is ample,Φ , n +2 ( r ⊕ ⊕ ⊕ , n +2 ( r ⊕ ⊕ ⊕ Φ , (1) ∼ MvN Φ , n +3 ( r ⊕ ⊕ ⊕ ∼ MvN Φ , n +3 ( r ⊕ ⊕ ⊕ ∼ MvN Φ , n +2 ( r ⊕ ⊕ ⊕ Φ , (0) ∼ MvN Φ , n +2 ( r ⊕ ⊕ ⊕ , n +2 ( r ⊕ ⊕
0) is ample. Considering that1 − Φ , n +2 ( r ⊕ ⊕
0) = Φ , n +2 ((1 − r ) ⊕ ⊕ ∼ MvN Φ , n +2 ((1 − r ) ⊕ ⊕ − Φ , n +2 ( r ⊕ ⊕
0) is also ample, and hence Φ , n +2 ( r ⊕ ⊕
0) is co-ample. (cid:3)
Lemma 4.6.
Let A be a strict unital C*-algebra that satisfies the Voiculescu property. If p, q ∈ Z A0 ( A ) are ampleand co-ample projections, then the following assertions are equivalent:(1) p, q represent the same element of K ( A ) ;(2) p, q are Murray–von Neumann equivalent;(3) p, q are unitary equivalent.Proof. The implications (3) ⇒ (2) ⇒ (1) hold in general.(1) ⇒ (3) Suppose that p, q ∈ Z A0 ( A ) are such that [ p ] = [ q ]. Then there exist n, k ∈ ω such that p ⊕ n ⊕ k ∼ MvN q ⊕ n ⊕ k .Since p, q are ample, we have p ⊕ n + k ∼ MvN p ⊕ n ⊕ k ∼ MvN q ⊕ n ⊕ k ∼ MvN q ⊕ n + k .Therefore, we have p ∼ MvN Φ n + k +1 ( p ⊕ n + k ) ∼ MvN Φ n + k +1 ( q ⊕ n + k ) ∼ MvN q . EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 31
Using the fact that p, q are co-ample, the same argument applied to 1 − p and 1 − q shows that 1 − p ∼ MvN − q .Thus, p, q are unitarily equivalent. (cid:3) Lemma 4.7.
Let A be a strict C*-algebra that satisfies the Voiculescu property. If p, q ∈ Z A0 ( A ) , then Φ , ( p ⊕ q ) ∈ Z A0 ( A ) .Proof. We need to show that Φ , ( p ⊕ q ) is ample and co-ample. We have thatΦ , ( p ⊕ q ) ⊕ , ( p ⊕ q ) ⊕ Φ , (1)= Φ , ( p ⊕ q ⊕ ∼ MvN Φ , ( p ⊕ q ⊕ ∼ MvN Φ , ( p ⊕ q ) ⊕ Φ , (0) ∼ MvN Φ , ( p ⊕ q ) ⊕ , ( p ⊕ q ) is ample. The same argument applied to 1 − ( p ⊕ q ) = (1 − p ) ⊕ (1 − q ) shows thatΦ , ( p ⊕ q ) is co-ample. (cid:3) Suppose that A is a strict unital C*-algebra satisfying Voiculescu’s property. Consider the unitary group U ( A ),which is a strictly closed subset of Ball ( A ) and hence a Polish group when endowed with the strict topology, and thestandard Borel space Z A0 ( A ) of projections in A that are both ample and co-ample, which by assumption is a Borelsubset of A invariant under unitary conjugation. We can consider the Borel action U ( A ) y Z A0 ( A ) by conjugation.We let B A0 ( A ) be the corresponding orbit equivalence relation, and K A0 ( A ) := Z A0 ( A ) (cid:14) B A0 ( A ) be the correspondingsemidefinable set. For p ∈ Z A0 ( A ), we let [ p ] B A0 ( A ) be the B A0 ( A )-class of p . The Borel functions ( p, q ) Φ , ( p ⊕ q )and p − p induce a semidefinable group structure on K A0 ( A ) with trivial element [Φ , (1 ⊕ B A0 ( A ) . SinceB A0 ( A ) is the orbit equivalence relation associated with a Borel action of a Polish group on B A0 ( A ), we have thatK A0 ( A ) is in fact a definable group by Corollary 1.16.The following proposition is an immediate consequence of the lemmas above. Proposition 4.8.
Suppose that A is a strict C*-algebra satisfying Voiculescu’s property. Adopt the notation fromDefinition 4.2. Then K A0 ( A ) and K ( A ) are definably isomorphic definable groups.Proof. By the above remarks, K A0 ( A ) is a definable group. Furthermore, the Borel functions Z A0 ( A ) → Z ( A ), p ( p,
0) and Z ( A ) → Z A0 ( A ), ( p, q ) Φ , n +2 ( p ⊕ (1 − q ) ⊕ ⊕
0) induce mutually inverse definable isomorphismsbetween K A0 ( A ) and K ( A ). By Lemma 1.11, this shows that K ( A ) is also a definable group, definably isomorphicto K A0 ( A ). (cid:3) Suppose that A is a separable, unital C*-algebra, and ρ : A → B ( H ) is a nondegenerate ample representation.Define the corresponding Paschke dual D ρ ( A ) as in Example 2.31 to be the algebra D ρ ( A ) = { T ∈ B ( H ) : ∀ a ∈ B ( H ) , T ρ ( a ) ≡ ρ ( a ) T mod K ( H ) } .Then, D ρ ( A ) is a strict unital C*-algebra, with respect to the strict topology on Ball ( D ρ ( a )) induced by theseminorms T max {k T S k , k ST k , k T ρ ( a ) − ρ ( a ) T k} for S ∈ K ( H ) and a ∈ A . We now observe that, as a consequence of Voiculescu’s theorem, D ρ ( A ) satisfies theVoiculescu property; see Definition 4.2.Let ρ n : A → B ( H n ) be the n -fold direct sum of ρ . Notice that, under the usual identification of B ( H n ) with M n ( B ( H )), D ρ n ( A ) corresponds to M n ( D ρ ( A )). For k, n ≥
1, as both ρ k and ρ n are ample representations of A ,by Voiculescu’s theorem there exists a surjective isometry V k,n : H k → H n such that Ad( V ) : B ( H n ) → B (cid:0) H k (cid:1) satisfies (Ad ( V ) ◦ ρ n ) ( a ) ≡ ρ k ( a ) mod K ( H ) for every a ∈ A . This implies that Ad( V ) induces a strict *-isomorphism Φ k,n := Ad ( V ) : M n ( D ρ ( A )) → M k ( D ρ ( A )). By Voiculescu’s theorem, Φ k,n does not depend, upto unitary equivalence, from the choice of the surjective isometry V k,n : H k → H n . Thus, we have that D ρ ( A )satisfies (2), (3), and (4) of Definition 4.2.Every projection P ∈ D ρ ( A ) defines a unital extension ϕ P : A → B ( P H ), a π ( P ρ ( a ) | P H ). By [HR00,Lemma 5.1.2], we have the following.
Lemma 4.9.
Suppose that
P, P , P ∈ D ρ ( A ) are projections. The following assertions are equivalent:(1) ϕ P , ϕ P are equivalent extensions;(2) P , P are Murray–von Neumann equivalent.Furthermore, the following assertions are equivalent:(1) P is ample;(2) ϕ P is injective. From Lemma 4.9 it is easy to deduce the following.
Proposition 4.10.
Suppose that A is a separable, unital C*-algebra, ρ : A → B ( H ) is a nondegenerate amplerepresentation of A , and D ρ ( A ) is the corresponding Paschke dual. Then D ρ ( A ) satisfies Voiculescu’s property.Proof. By Lemma 4.9, a projection P ∈ D ρ ( A ) is ample if and only if ϕ P is injective. This is equivalent to theassertion that, for every self-adjoint a ∈ A , every S ∈ K ( H ), and every ε > k P ρ ( a ) − S k > k a k − ε .By strict lower semicontinuity of the norm in D ρ ( A ), this is an open condition. This shows that the set of ampleprojections is a G δ set.Since ρ is an ample representation, and ϕ I = ρ , we have that I ∈ D ρ ( A ) is ample.Finally, we need to verify (1) of Definition 4.2. For n > k , and projection P ∈ M k ( D ρ ( A )) = D ρ k ( A ) we have Q := Φ k,n ( P ⊕ n − k ) = V ∗ k,n ( P ⊕ n − k ) V k,n .Thus, ϕ Q is equivalent to ϕ P ⊕ n − k = ϕ P . Hence, by Lemma 4.9, Q and P are Murray–von Neumann equivalent. (cid:3) As a consequence of Proposition 4.10 and Proposition 4.8 we have the following.
Proposition 4.11.
Suppose that A is a separable C*-algebra, and ρ : A → B ( H ) is a nondegenerate amplerepresentation of A . Then K ( D ρ ( A )) and K A0 ( D ρ ( A )) are definably isomorphic definable groups. Suppose that
A, B are separable unital C*-algebras. Recall that, if ρ, ρ ′ are linear maps from A to B ( H ), and V : H → H is an isometry, then we write ρ ′ . V ρ if ρ ′ ( a ) ≡ V ∗ ρ ( a ) V mod K ( H ) for every a ∈ A . Suppose that A, B are separable unital C*-algebras, α : A → B is a unital *-homomorphism. Let ρ A , ρ B be ample representationsof A, B on a Hilbert space H . An isometry V α : H → H covers α if ρ A . V α ρ B ◦ α .We have that for every unital *-homomorphism α : A → B there exists an isometry V α : H → H that covers α [HR00, Lemma 5.2.3]. Furthermore, Ad ( V α ) induces a strict unital *-homomorphism Ad ( V α ) : D ρ B ( B ) → D ρ A ( A ).In turn, Ad ( V α ) : D ρ B ( B ) → D ρ A ( A ) induces a definable group homomorphism K ( D ρ B ( B )) → K ( D ρ A ( A )).This definable group isomorphism only depends on α , and not on the choice of the isometry V α that covers α [HR00, Lemma 5.2.4]. This gives a contravariant functor A K ( D ρ A ( A )) from the category of separable unitalC*-algebras to the category of definable groups. Similarly, one can regard A K A0 ( D ρ A ( A )) as a contravariantfunctor, naturally isomorphic to K ( D ρ A ( A )).Using Proposition 4.11 we can show the following. Proposition 4.12.
Suppose that A is a separable, unital C*-algebra, and ρ : A → B ( H ) is a nondegenerate amplerepresentation of A . Then Ext ( A ) − is a definable group, which is naturally definably isomorphic to K ( D ρ ( A )) .Proof. Consider the corresponding Paschke dual D ρ ( A ). Recall that E ( A ) denotes the Polish space of representa-tives of injective, unital, semi-split extensions of A by K ( H ), which are ample ucp maps ϕ : A → B ( H ) satisfying ϕ ( xy ) ≡ ϕ ( x ) ϕ ( y ) mod K ( H ) for x, y ∈ A . An ample and co-ample projection P ∈ D ρ ( A ) determines an exten-sion ϕ P ∈ E ( A ) of A , defined as follows. Choose a linear isometry V : H → H such that V V ∗ = P , and define ϕ P ( a ) = V ∗ ρ ( a ) V . (Notice that V can be chosen in a Borel fashion from P .) The Borel function P ϕ P inducesa definable group isomorphism γ : K A0 ( D ρ ( A )) → Ext ( A ) − ;see [HR00, Proposition 5.1.6]. EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 33
We claim that the inverse group homomorphism γ − : Ext ( A ) − → K A0 ( D ρ ( A )) is definable as well. Indeed, if ϕ ∈ E ( A ) then, by the Definable Voiculescu Theorem (Lemma 2.42), one can choose in a Borel way an isometry V ϕ : H → H such that ϕ . V ϕ ρ . Thus, we have that ϕ ( a ) ≡ V ∗ ϕ ρ ( a ) V ϕ mod K ( H )for every a ∈ A . If P := V ϕ V ∗ ϕ then we have that P is a projection in D ρ ( A ) such that ϕ P is equivalent to ϕ . As P is not necessarily ample and co-ample, one can replace P with Φ , ( P ⊕ ⊕
0) to obtain an ample and co-ampleprojection P ϕ ∈ Z ( D ρ ( A )) such that ϕ P ϕ is equivalent to ϕ . Thus the Borel function ϕ P ϕ is a lift of the inversemap γ − : Ext ( A ) − → K A0 ( D ρ ( A )). This shows that γ − is also definable. Therefore, γ is a natural isomorphismin the category of semidefinable groups.As K A0 ( D ρ ( A )) is in fact a definable group, this implies that Ext ( A ) − is a definable group. Since K A0 ( D ρ ( A ))is naturally definably isomorphic to K ( D ρ ( A )), we have that Ext ( A ) − is naturally definably isomorphic toK ( D ρ ( A )) as well. (cid:3) Definable K -theory of commutants in the Calkin algebra . Suppose that A is a unital separable C*-algebra, and ρ is an ample representation of A on the infinite-dimensional separable Hilbert space H . Then ρ induces an ample representation ρ + of the unitization A + on H ⊕ H , defined by ρ + ( a ) = ρ ( a ) ⊕ a ∈ A . Recallthat the Paschke dual algebra is the strict unital C*-algebra D ρ ( A ) := { T ∈ B ( H ) : ∀ a ∈ A, T ρ ( a ) ≡ ρ ( a ) T mod K ( H ) } .We also have the Paschke dual algebra D ρ + (cid:0) A + (cid:1) = (cid:8) T ∈ B ( H ⊕ H ) : ∀ a ∈ A, T ρ + ( a ) ≡ ρ + ( a ) T mod K ( H ⊕ H ) (cid:9) .Notice that D ρ + (cid:0) A + (cid:1) = (cid:20) D ρ ( A ) K ( H ) K ( H ) B ( H ) (cid:21) ;see [HR00, Section 5.2].Define J to be the strict ideal (cid:20) K ( H ) K ( H ) K ( H ) B ( H ) (cid:21) of D ρ + ( A + ). Let also D ρ + ( A//A ) be the strict ideal (cid:8) T ∈ D ρ + (cid:0) A + (cid:1) : ∀ a ∈ A, T ρ + ( a ) ≡ K ( H ⊕ H ) (cid:9) of D ρ + ( A + ); see Proposition 2.30. Lemma 4.13.
The C*-algebras J and D ρ + ( A//A ) defined above have trivial K -theory.Proof. The assertion about J follows by considering the six-term exact sequence in K-theory associated with the pair( J , M ( K ( H ))); see also [HR00, Exercise 4.10.9]. The assertion about D ρ + ( A//A ) is [HR00, Lemma 5.4.1]. (cid:3)
Lemma 4.14.
Suppose that A is a separable unital C*-algebra. For i ∈ { , } :(1) the definable group homomorphism K i (cid:0) D ρ + ( A + ) (cid:1) → K i (cid:0) D ρ + ( A + ) / J (cid:1) is an isomorphism in the categoryof semidefinable groups;(2) the strict *-homomorphism ϕ : D ρ ( A ) → D ρ + ( A + ) , x x ⊕ induces an isomorphism K i ( D ρ ( A ) /K ( H )) → K i (cid:0) D ρ + (cid:0) A + (cid:1) / J (cid:1) in the category of semidefinable groups.(3) The map K ( D ρ ( A )) → K ( D ρ ( A ) /K ( H )) is an isomorphism in the category of semidefinable groups;(4) The subgroup G of K ( D ρ ( A ) /K ( H )) , consisting of the kernel of the (surjective) index map ∂ : K ( D ρ ( A ) /K ( H )) → K ( K ( H )) ∼ = Z is Borel, and the definable group homomorphism K ( D ρ ( A )) → K ( D ρ ( A ) /K ( H )) induces an isomor-phism K ( D ρ ( A )) → G in the category of semidefinable groups. Proof. (1) Since J has trivial K-theory, the group homomorphism K i (cid:0) D ρ + ( A + ) (cid:1) → K i (cid:0) D ρ + ( A + ) / J (cid:1) is an isomor-phism. We need to prove that the inverse group homomorphism K i (cid:0) D ρ + ( A + ) / J (cid:1) → K i (cid:0) D ρ + ( A + ) (cid:1) is definable.Consider first the case i = 0. Consider p ∈ Z (cid:0) D ρ + ( A + ) / J (cid:1) . Thus, p ∈ Proj (cid:0) M d (cid:0) D ρ + ( A ) (cid:1) /M d ( J ) (cid:1) for some d ≥
1. After replacing ρ with ρ d we can assume that d = 1. Thus, p ∈ D ρ + ( A ) is a mod J projection. This impliesthat p = (cid:20) p p p p (cid:21) where p ∈ D ρ ( A ) is a mod K ( H ) projection. Then by Lemma 2.50, one can choose in a Borel fashion from p aprojection q ∈ D ρ ( A ) such that q ≡ p mod K ( H ) and hence q ⊕ ≡ p mod J .We now consider the case when i = 1. Consider q ∈ Z (cid:0) D ρ + ( A + ) / J (cid:1) . Thus, q ∈ U (cid:0) M d (cid:0) D ρ + ( A ) (cid:1) /M d ( J ) (cid:1) forsome d ≥
1. After replacing ρ with ρ d we can assume that d = 1. Thus, u = (cid:20) u u u u (cid:21) where u ∈ D ρ + ( A ) is a mod K ( H ) unitary. Let v ∈ D ρ ( A ) be the partial isometry in the polar decompositionof u , which depends in a Borel fashion from u by Lemma 2.53. Then we have that v := (cid:20) v I − vv ∗ I − v ∗ v v ∗ (cid:21) ∈ D ρ + (cid:0) A + (cid:1) is a unitary such that v ≡ u mod J .(2) Since ϕ induces a *-isomorphism D ρ ( A ) /K ( H ) → D ρ + ( A + ) / J , it induces a definable group isomorphismK i ( D ρ ( A ) /K ( H )) → K i (cid:0) D ρ + ( A + ) / J (cid:1) . It is immediate to verify that the inverse group homomorphism is alsodefinable, as it is induced by the Borel function (cid:20) x x x x (cid:21) x .(3) and (4): Under the isomorphism K ( K ( H )) ∼ = Z , the definable group homomorphism K ( D ρ ( A ) /K ( H )) → K ( K ( H )) ∼ = Z maps each T ∈ U ( D ρ ( A ) /K ( H )) to its Fredholm index, and in particular it is surjective. As theFredholm index is given by a Borel map, and K ( K ( H )) = { } , it follows from the six-term exact sequence in K-theory associated with D ρ ( A ) and K ( H ) that K ( D ρ ( A )) → K ( D ρ ( A ) /K ( H )) is a definable group isomorphism,and that K ( D ρ ( A )) → K ( D ρ ( A ) /K ( H )) is an injective definable group homomorphism with range equal to G . The inverse K ( D ρ ( A ) /K ( H )) → K ( D ρ ( A )) is definable by Lemma 2.50. The inverse G → K ( D ρ ( A )) isdefinable by Lemma 2.53, considering that given T ∈ U ( B ( H ) /K ( H )) such that index ( T ) = 0, then the partialisometry U in the polar decomposition of T is a unitary such that U ≡ T mod K ( H ). (cid:3) Corollary 4.15.
Suppose that A is a separable unital C*-algebra, and ρ is an ample representation of A . Then Ext ( A ) − , K ( D ρ ( A )) , K ( D ρ ( A ) /K ( H )) , K (cid:0) D ρ + ( A + ) / J (cid:1) , and K (cid:0) D ρ + ( A + ) (cid:1) are definably isomorphic de-finable groups.Proof. This is a consequence of Lemma 4.14, Proposition 4.12, and Corollary 1.14. (cid:3)
Suppose that A is a unital separable C*-algebra. Let C ( T , A ) be the unital separable C*-algebra of continuousfunctions f : T → A . We identify A with the C*-subalgebra of C ( T , A ) consisting of constant functions. The suspension SA of A is the C*-subalgebra { f ∈ C ( T , A ) : f (1) = 0 } . The unital suspension Σ A of A is the unitizationof SA , which can be identified with { f ∈ C ( T , A ) : f (1) ∈ C } .Consider an ample representation ρ of C ( T , A ) on a Hilbert space H , and let ρ Σ A be its restriction to Σ A and ρ A be its restriction to A . We can then consider the Paschke dual algebras D ρ Σ A (Σ A ) ⊆ B ( H ) and D ρ A ( A ) ⊆ B ( H ). Lemma 4.16.
Suppose that A is a separable unital C*-algebra. Then K ( D ρ A ( A ) /K ( H )) is a definable group,definably isomorphic to the definable group K ( D ρ Σ A (Σ A ) /K ( H )) .Proof. A definable group isomorphismK ( D ρ Σ A (Σ A ) /K ( H )) → K ( D ρ A ( A ) /K ( H )) EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 35 is described in [Pas81, Theorem 6], as follows. Let U ∈ C ( T , A ) be the function λ λ
1. Let p be a mod K ( H )projection in D ρ Σ A (Σ A ). Then f ( p ) := pU + (1 − p ) ∈ D ρ A ( A ) is a mod K ( H ) unitary. A similar definition formod K ( H ) projections over D ρ Σ A (Σ A ) defines a Borel function Z ( D ρ Σ A (Σ A ) /K ( H )) → Z ( D ρ A ( A ) /K ( H )), p f ( p ). It is proved in [Pas81, Theorem 6] that this Borel function induces an isomorphism K ( D ρ Σ A (Σ A ) /K ( H )) → K ( D ρ A ( A ) /K ( H )).By Corollary 4.15 we have that K ( D ρ Σ A (Σ A ) /K ( H )) is a definable group. Thus, by Proposition 1.14 we havethat K ( D ρ A ( A ) /K ( H )) is a definable group as well. (cid:3) Proposition 4.17.
Suppose that A is a separable unital C*-algebra, and ρ is an ample representation of A . Then Ext (Σ A ) − , K ( D ρ ( A ) /K ( H )) , K (cid:0) D ρ + ( A + ) / J (cid:1) , and K (cid:0) D ρ + ( A + ) (cid:1) are definably isomorphic definable groups.Proof. Let ρ Σ A and ρ A be the ample representations of Σ A and A , respectively, as in Lemma 4.16. Then byCorollary 4.15, Ext (Σ A ) − and K ( D ρ Σ A (Σ A ) /K ( H )) are definably isomorphic definable groups. By Lemma4.16, K ( D ρ Σ A (Σ A ) /K ( H )) and K ( D ρ A ( A ) /K ( H )) are definably isomorphic definable groups. By Voiculescu’stheorem, D ρ A ( A ) /K ( H ) and D ρ ( A ) /K ( H ) are isomorphic in the category of unital C*-algebras with a strictcover; see Lemma 2.46. In particular, K ( D ρ A ( A ) /K ( H )) and K ( D ρ ( A ) /K ( H )) are isomorphic in the categoryof semidefinable groups. From this and Corollary 1.14, it follows that K ( D ρ ( A ) /K ( H )) is a definable group.Finally, K (cid:0) D ρ + ( A + ) / J (cid:1) and K (cid:0) D ρ + ( A + ) (cid:1) are definable groups, definably isomorphic to K ( D ρ ( A ) /K ( H )) byLemma 4.14 and Corollary 1.14 again. (cid:3) Definable K -homology. Suppose that A is a separable C*-algebra. Fix an ample representation ρ + of A + ,and define D ( A ) := D ρ + ( A + ). The K-homology groups of A are the definable groupsK ( A ) := K ( D ( A )) ∼ = Ext (cid:0) A + (cid:1) − and K ( A ) := K ( D ( A )) ∼ = Ext(( SA ) + ) − ;see [HR00, Definition 5.2.7]. By Proposition 4.12, K p ( − ) for p ∈ { , } is a contravariant functor from the categoryof separable C*-algebras to the category of definable abelian groups.When A is a separable unital C*-algebra, one can also define the reduced
K-homology groups by considering anample representation ρ of A and the corresponding Pachke dual algebra ˜ D ( A ) := D ρ ( A ) and set˜K ( A ) := K ( ˜ D ( A )) ∼ = Ext ( A ) − and ˜K ( A ) := K ( ˜ D ( A ));see [HR00, Definition 5.2.1].Suppose now that A is a separable C*-algebra, and J is a closed two-sided ideal of A . Fix as above an amplerepresentation ρ + of A + . Define D ( A ) := D ρ + ( A + ) as above, and set D ( A//J ) to be the strict ideal (cid:8) T ∈ D ( A ) : ∀ a ∈ J, T ρ + ( a ) ≡ K ( H ) (cid:9) of D ( A ). Lemma 4.18.
Suppose that A is a separable C*-algebra, and i ∈ { , } . Then K i ( D ( A ) / D ( A//A )) is a definablegroup, definably isomorphic to K − i ( A ) .Proof. By Lemma 4.13, D ( A//A ) has trivial K-theory. Thus, by the six-term exact sequence in K-theory, thedefinable group homomorphism K i ( D ( A + )) → K i ( D ( A + ) / D ( A//A )) is an isomorphism. Since K i ( D ( A + )) is adefinable group by Proposition 4.17 and Corollary 4.15, the conclusion follows from Corollary 1.14. (cid:3) Lemma 4.19.
Suppose that A is a separable C*-algebra, and J is a closed two-sided ideal of A . Then the inclusionmap D ( A ) ⊆ D ( J + ) induces an isomorphism D ( A ) / D ( A//J ) → D ( J ) / D ( J//J ) in the category of separableunital C*-algebras with a strict cover. Proof.
We identify A + with its image inside B ( H ) under ρ + . It follows from the definition that D ( A//J ) = D ( J//J ) ∩ D ( A ). Thus, the inclusion map D ( A ) ⊆ D ( J ) induces a definable injective unital *-homomorphism D ( A ) / D ( A//J ) → D ( J ) / D ( J//J ), which is in fact onto [HR00, Theorem 5.4.5]. It remains to prove that theinverse unital *-isomorphism D ( J ) / D ( J//J ) → D ( A ) / D ( A//J ) is also definable. This amounts at noticing thatthe proof of [HR00, Theorem 5.4.5] via Kasparov’s Technical Theorem [HR00, Theorem 3.8.1] can be used todescribe a Borel lift D ( J ) → D ( A ) of the unital *-isomorphism D ( J ) / D ( J//J ) → D ( A ) / D ( A//J ).For T ∈ D ( J ) let E ( T ) be closed linear span of { [ a, T ] , [ a, T ∗ ] : a ∈ A } . Fix a dense sequence ( j m ) in Ball ( J ),a dense sequence ( a m ) in Ball ( A ), and a dense sequence ( b m ) in Ball ( K ( H )). Notice that, for j ∈ J , and a ∈ A ,and T ∈ D ( J ), we have that j [ a, T ] = jaT − jT a ≡ jaT − T ja ≡ K ( H ) .Fix an approximate unit ( u n ) for J such that, for m ≤ n , k u n j m − j m k ≤ − n and k u n a m − a m u n k ≤ − n . Fix an approximate unit ( w n ) n ∈ ω for K ( H ) such that, if we set d n := ( w n − w n − ) / ,then we have, for m ≤ n , k d n b m k ≤ − n k d n j m − j m d n k ≤ − n k d n a m − a m d n k ≤ − n .One can see that such an approximate unit for K ( H ) exists by considering a approximate unit for K ( H ) that isquasicentral for J and A [HR00, Theorem 3.2.6] and then a suitable subsequence via a diagonal argument.Fix T ∈ D ( J ). Then using the Lusin–Novikov Selection Theorem [Kec95, Theorem 18.10] and [HR00, Theorem3.2.6] one can see that one can recursively define, for n ∈ ω , ℓ Tn ∈ ω , k Tn, , . . . , k Tn,ℓ Tn ≥ n , and t Tn, , . . . , t Tn,ℓ Tn ∈ [0 , ∩ Q that depend in a Borel fashion from T such that, setting w Tn := t Tn, w k Tn, + · · · + t Tn,ℓ Tn w k Tn,ℓTn and d Tn := (cid:0) w Tn − w Tn − (cid:1) / one has that w Tn , d Tn ∈ K ( H ) depend in a Borel fashion from T and, for m , m , m ≤ n , (cid:13)(cid:13) d Tn [ a m , T ] − [ a m , T ] d Tn (cid:13)(cid:13) ≤ − n (cid:13)(cid:13) d Tn [ T, j m ] (cid:13)(cid:13) ≤ − n (cid:13)(cid:13) d Tn u m [ a m , T ] (cid:13)(cid:13) ≤ − n .Furthermore, we also have from the choice of ( w n ) that, for m ≤ n , (cid:13)(cid:13) d Tn b m (cid:13)(cid:13) ≤ − n (cid:13)(cid:13) d Tn j m − j m d Tn (cid:13)(cid:13) ≤ − n (cid:13)(cid:13) d Tn a m − a m d Tn (cid:13)(cid:13) ≤ − n .As in the proof of Kasparov’s Technical Theorem [HR00, Theorem 3.8.1], one has that X n ∈ ω d Tn u n d Tn converges in the strong-* topology to some positive element X T ∈ Ball ( B ( H )). Furthermore, we have that(1 − X T ) j ≡ K ( H ) X T [ T, a ] ≡ K ( H )[ X T , a ] ≡ K ( H ) EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 37 for j ∈ J and a ∈ A . Thus, X T T ∈ D ( A ) and (1 − X T ) T ∈ D ( J//J ). Indeed, if a ∈ A then we have that[ X T T, a ] = X T T a − aX T T = X T T a − X T aT + X T aT − aX T T = X T [ T, a ] + [ X T , a ] T ≡ K ( H ) .If j ∈ J then we have that (1 − X T ) T j ≡ (1 − X T ) jT ≡ K ( H ) .We have that the function D ( J ) K ( H ), T [ X T , a ] is Borel, being the pointwise limit of Borel functions T n X k =0 (cid:2) d Tk u k d Tk , a (cid:3) for n ∈ ω . Thus, the function D ( J ) K ( H ), T [ X T , a ] T is Borel as well. For the same reasons, thefunction D ( J ) K ( H ), T X T [ T, a ] is Borel, and hence the function D ( J ) K ( H ), T [ X T T, a ] = X T [ T, a ] + [ X T , a ] T is Borel. A similar argument shows that the function D ( J ) → K ( H ), T X T T b is Borel for b ∈ K ( H ). Therefore, the function D ( J ) → D ( A ), T X T T is Borel. Since T − X T T = (1 − X T ) T ∈ D ( J//J ),we have that T X T T is a lift of the unital *-isomorphism D ( J ) / D ( J//J ) → D ( A ) / D ( A//J ). This concludesthe proof. (cid:3)
Corollary 4.20.
Suppose that A is a separable C*-algebra, and J is a closed two-sided ideal of A . Fix i ∈ { , } .Then K i ( D ( A + ) / D ( A//J )) is a definable group, definably isomorphic to K i ( D ( J + ) / D ( J//J )) .Proof. By Lemma 4.18, K i ( D ( J + ) / D ( J//J )) is a definable group. By Lemma 4.19, K i ( D ( A + ) / D ( A//J )) isisomorphic to K i ( D ( J + ) / D ( J//J )) in the category of semidefinable groups. Whence, the conclusion follows fromLemma 1.11. (cid:3)
Suppose as above that A is a separable C*-algebra, and J is a closed two-sided ideal of A . One defines for i ∈ { , } the relative K-homology groupsK i ( A, A/J ) := K − i ( D ( A ) / D ( A//J )) ;see [HR00, Definition 5.3.4]. These are definable groups by Corollary 4.20. The assignment (
A, J ) K i ( A, A/J )gives a contravariant functor from the category of separable C*-pairs to the category of definable groups. Here,a separable C*-pair is a pair (
A, I ) where A is a separable C*-algebra and I is a closed two-sided ideal of A . Amorphism ( A, I ) → ( B, J ) of separable C*-pairs is a *-homomorphism A → B that maps I to J . If α : ( A, I ) → ( B, J ) is a morphism of C*-pairs, and V : H → H is an isometry that covers α + : A + → B + , then we havethat the corresponding strict unital *-homomorphism Ad ( V ) : D ( B + ) → D ( A + ) maps D ( B//J ) to D ( A//I ).Thus, it induces a definable unital *-homomorphism D ( B ) / D ( B//J ) → D ( A ) / D ( A//I ), and a definable grouphomomorphisms K i ( B, B/J ) → K i ( A, A/I ).Suppose that (
A, J ) is a separable C*-pair. The natural definable isomorphismsK i ( A, A/J ) = K − i (cid:0) D (cid:0) A + (cid:1) / D ( A//J ) (cid:1) ∼ = K − i ( D ( J ) / D ( J//J )) ∼ = K − i ( D ( J )) = K i ( J )from Lemma 4.19 and Lemma 4.14 give a natural definable isomorphism K i ( A, A/J ) ∼ = K i ( J ) called the excisionisomorphism ; see [HR00, Theorem 5.4.5].Suppose that ( A, J ) is a separable C*-pair. We say that (
A, J ) is semi-split if the short exact sequence0 → J → A → A/J → A + → A + /J admits a ucp rightinverse. By the Choi–Effros lifting theorem [CE76], every nuclear separable C*-pair is semi-split. Suppose that( A, J ) is semisplit. If V : H → H is a linear isometry that covers the quotient map A → A/J , then the unital*-homomorphism Ad ( V ) : D ( A/J ) → D ( A//J ) induces a natural definable isomorphism in K-theory [HR00,Proposition 5.3.7]. In this case, from the six-term exact sequence in K-theory K ( D ( A//J )) K ( D ( A )) K ( D ( A ) / D ( A//J ))K ( D ( A ) / D ( A//J )) K ( D ( A )) K ( D ( A//J ))associated with the strict unital C*-pair ( D ( A ) , D ( A//J )), one obtains the six-term exact sequence in K-homologyassociated with the separable C*-pair (
A, J )K ( A/J ) K ( A ) K ( A, A/J )K ( A, A/J ) K ( A ) K ( A/J )as in [HR00, Theorem 5.3.10], where the connecting maps are definable homomorphisms.5.
The Kasparov and Cuntz pictures of definable K -homology In this section we recall the notion of graded Hilbert space and of (graded) Fredholm module for a separableC*-algebra as in [HR00, Chapter 8 and Appendix A]. We also recall Kasparov’s description of K-homology groupsin terms of Fredholm modules from [Kas75]. We then show that Kasparov’s K-homology groups can be regardedas definable groups, and are definably isomorphic to the K-homology groups as defined in the previous section.We conclude by recalling the Cuntz picture for K-homology from [Cun87]; see also [Hig87] and [JT91, Chapter 5].Again, we show that the Cuntz K-homology groups can be seen as definable groups, and are naturally definablyisomorphic to the K-homology groups as previously defined.5.1.
Graded vector spaces and algebras.
Let V be a vector space. A grading of V is a decomposition V = V + ⊕ V − as a direct sum of two subspaces, called the positive and negative part of V . The corresponding gradingoperator γ V is the involution of V whose eigenspaces for 1 and − V + and V − , respectively. A vector spaceendowed with a grading is a graded vector space . The opposite of the graded vector space V is the graded vectorspace V op obtained from V by interchanging the positive and the negative part. An endomorphism T of V is even if T ( V + ) ⊆ V + and T ( V − ) ⊆ V − or, equivalently, γ V T = T γ V ; it is odd if T ( V + ) ⊆ V − and T ( V − ) ⊆ V + or,equivalently, γ V T = − T γ V .A graded Hilbert space is Hilbert space endowed with a grading whose positive and negative parts are closedorthogonal subspaces or, equivalently, the grading operator is a self-adjoint unitary.A graded algebra is a complex algebra that is also a graded vector space, and such that: A + · A + ∪ A − · A − ⊆ A + and A + · A − ∪ A − · A + ⊆ A − or, equivalently, the grading operator γ A is an algebra automorphism of A . The elements of A + are even elementsof the algebra, and the elements of A − are called odd elements of the algebra. An element is homogeneous if it iseither even or odd. The degree ∂a of an even element a is 0, while the degree ∂a of an odd element a is 1. Thegraded commutator of elements of A is defined for homogeneous elements by[ a, a ′ ] = aa ′ − ( − ∂a∂a ′ a ′ a and extended by linearity.A graded C*-algebra A is a C*-algebra that is also a graded algebra and such that A + and A − are closedself-adjoint subspaces or, equivalently, the grading operator γ A is a C*-algebra automorphism of A . Example 5.1.
Suppose that V is a graded vector space. Then the algebra End ( V ) of endomorphisms of V isa graded algebra, with End ( V ) + equal to the set of even endomorphisms of V , and End ( V ) − is the set of oddendomorphisms of V .If H is a graded Hilbert space, then B ( H ) ⊆ End ( H ) is a graded C*-algebra. EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 39
Example 5.2.
Fix n ≥
1. Define C n to be the graded complex unital *-algebra generated by n odd operators e , . . . , e n such that, for distinct i, j ∈ { , , . . . , n } , e i e j + e j e i = 0, e j = − e ∗ j = − e j .As a complex vector space, C n has dimension 2 n , where monomials e i · · · e i k for 1 ≤ i < · · · < i k ≤ n and0 ≤ k ≤ n comprise a basis. Declaring these monomials to be orthogonal defines an inner product on C n . The leftregular representation of C n on the Hilbert space C n turns C n into a graded C*-algebra.Suppose that V and V are graded vector spaces. The graded tensor product V := V ˆ ⊗ V is the tensor productof V and V equipped with the grading operator γ V := γ V ⊗ γ V . Thus, we have that V + = (cid:0) V +1 ⊗ V +2 (cid:1) ⊕ (cid:0) V − ⊗ V − (cid:1) V − = (cid:0) V +1 ⊗ V − (cid:1) ⊕ (cid:0) V − ⊗ V +2 (cid:1) .If A and A are graded algebras, then the graded tensor product A := A ˆ ⊗ A (as graded vector spaces) is a gradedalgebra with respect to the multiplication operation defined on homogeneous elementary tensors by (cid:0) a ˆ ⊗ a (cid:1) (cid:0) a ′ ˆ ⊗ a ′ (cid:1) = ( − ∂a ∂a ′ (cid:0) a a ′ ˆ ⊗ a a ′ (cid:1) .When V , V are graded vector spaces, then there is canonical inclusionEnd ( V ) ˆ ⊗ End ( V ) ⊆ End (cid:0) V ˆ ⊗ V (cid:1) obtained by setting, for homogeneous T i ∈ End ( V i ) and v i ∈ V i , (cid:0) T ˆ ⊗ T (cid:1) (cid:0) v ˆ ⊗ v (cid:1) = ( − ∂v ∂T ( T v ⊗ T v ) .We have that End ( V ) ˆ ⊗ End ( V ) = End (cid:0) V ˆ ⊗ V (cid:1) when V , V are finite-dimensional. Example 5.3.
There is a canonical isomorphism C m ˆ ⊗ C n ∼ = C m + n , obtained by mapping e i ˆ ⊗ e i and 1 ˆ ⊗ e j to e m + j for i ∈ { , , . . . , m } and j ∈ { , , . . . , n } .Fix p ≥
0. A p -graded Hilbert space is a graded Hilbert space endowed with p odd operators ε , . . . , ε p such that ε i ε j + ε j ε i = 0, ε j = −
1, and ε ∗ j = − ε j for distinct i, j ∈ { , , . . . , p } . Equivalently, a p -graded Hilbert space canbe thought of as a graded right module over C p , where one sets xe i := ε i ( x )for i ∈ { , , . . . , p } and x ∈ H . A 0-graded Hilbert space is simply a graded Hilbert space. By convention, a( − H and H are p -graded Hilbert spaces. A p -graded bounded linear map H → H is a bounded linear map that is also a right C p -module map. Example 5.4.
Suppose that
H, H ′ are p -graded Hilbert space. Then H op is p -graded, where ε H op i = − ε Hi for1 ≤ i ≤ n , and H ⊕ H ′ is p -graded, where ε H ⊕ H ′ i = ε Hi ⊕ ε H ′ i . Example 5.5. If H is p -graded and H is p -graded, then considering the isomorphism C p ˆ ⊗ C p ∼ = C p + p , andthe inclusion B (cid:0) H ˆ ⊗ H (cid:1) ⊆ B ( H ) ˆ ⊗ B ( H ), we have that H ˆ ⊗ H is ( p + p )-graded.A straightforward verification shows the following; see [HR00, Proposition A.3.4]. Proposition 5.6.
For p ≥ , the categories of p -multigraded and ( p + 2) -multigraded Hilbert spaces are equivalent.The categories of Hilbert spaces and linear maps and -graded Hilbert spaces and even 1 -graded linear maps areequivalent. Fredholm modules.
Suppose that A is a separable C*-algebra. We now recall the definition of Fredholmmodule over A ; see [HR00, Definition 8.1.1]. For each dimension d ∈ ω ∪ {ℵ } fix a Hilbert space H d of dimension d . Definition 5.7.
Suppose that A is a separable C*-algebra. A Fredholm module over A is a triple ( F, ρ, H ) suchthat: • H is a separable Hilbert space H d for some d ∈ ω ∪ {ℵ } ; • ρ : A → B ( H ) is a *-homomorphism; • F ∈ B ( H ) satisfies (cid:0) F − (cid:1) ρ ( a ) ≡ ( F − F ∗ ) ρ ( a ) ≡ [ F, ρ ( a )] ≡ K ( H ) for every a ∈ A . Remark 5.8.
Recall that, if H is a separable Hilbert space, then B ( H ) is a standard Borel space with respect tothe Borel structure induced by the strong-* topology on Ball ( B ( H )). Similarly, the Banach space L ( A, B ( H )) ofbounded linear maps A → B ( H ) is a standard Borel space when endowed with the Borel structure induced by thetopology of pointwise strong-* convergence on Ball ( L ( A, B ( H ))). The set F − ( A ) of Fredholm modules over A can thus be naturally regarded as a standard Borel space.The definition of graded Fredholm module is similar, where one replaces Hilbert spaces with graded
Hilbertspaces.
Definition 5.9.
Suppose that A is a separable C*-algebra. A graded Fredholm module over A is a triple ( F, ρ, H )such that: • H is a separable graded Hilbert space of the form ( H d , γ ) for some d ∈ ω ∪ {ℵ } and some grading operator γ on H d ; • ρ : A → B ( H ) is a *-homomorphism such that, for every a ∈ A , ρ ( a ) ∈ B ( H ) + is even , and hence ρ = ρ + ⊕ ρ − for some representations ρ ± of A on H ± , where we regard B ( H ) as a graded C*-algebra; • F ∈ B ( H ) is an odd operator that satisfies, for every a ∈ A , (cid:0) F − (cid:1) ρ ( a ) ≡ ( F − F ∗ ) ρ ( a ) ≡ [ F, ρ ( a )] ≡ K ( H ). Remark 5.10.
Again, we have that the set F ( A ) of Fredholm modules over A is a standard Borel space.The notions of graded and ungraded Fredholm modules can be recognized as particular instances (for p = 0 and p = −
1, respectively) of the notion of p -graded Fredholm module; see [HR00, Definition 8.1.11]. Definition 5.11.
Suppose that A is a separable C*-algebra. A p -multigraded Fredholm module over A is a triple( F, ρ, H ) such that: • H is a separable p -multigraded Hilbert space H of the form ( H d , γ, ε , . . . , ε p ) for d ∈ ω ∪ {ℵ } , gradingoperator γ on H d , and odd operators ε , . . . , ε d on ( H d , γ ); • a *-homomorphism ρ : A → B ( H ) such that, for every a ∈ A , ρ ( a ) is an even p - multigraded operator on H ; • F ∈ B ( H ) is an odd p - multigraded operator on H such that, for every a ∈ A , (cid:0) F − (cid:1) ρ ( a ) ≡ ( F − F ∗ ) ρ ( a ) ≡ [ F, ρ ( a )] ≡ K ( H ). Remark 5.12.
As in the cases p = 0 and p = −
1, the set F p ( A ) of p -multigraded Fredholm modules over A is astandard Borel space.We recall the notion of degenerate p -multigraded Fredholm module; see [HR00, Definition 8.2.7]. Definition 5.13.
Suppose that A is a separable C*-algebra. A p -multigraded Fredholm module ( F, ρ, H ) over A is degenerate if (cid:0) F − (cid:1) ρ ( a ) = ( F − F ∗ ) ρ ( a ) = [ F, ρ ( a )] for every a ∈ A .It is clear that the set D p ( A ) of degenerate p -multigraded Fredholm modules is a Borel subset of F p ( A ).Given p -multigraded Fredholm modules x = ( ρ, H, F ) and x ′ = ( ρ ′ , H ′ , F ′ ) over A , their sum is the p -multigradedFredholm module x ⊕ x ′ = ( ρ ⊕ ρ ′ , H ⊕ H ′ , F ⊕ F ′ ). The opposite of x is the p -multigraded Fredholm module x op = ( ρ, H op , − F ). The sum and opposite define Borel functions F p ( A ) × F p ( A ) → F p ( A ), ( x, x ′ ) x ⊕ x ′ andF p ( A ) → F p ( A ), x x op .Let A be a separable C*-algebra, and fix p ≥ −
1. Suppose that ( ρ, H, F ) and ( ρ ′ , H ′ , F ′ ) are p -multigradedFredholm modules. Then ( ρ, H, F ) and ( ρ, H ′ , F ′ ) are: EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 41 • unitarily equivalent if there exists an even p -multigraded unitary linear map U : H ′ → H such that F ′ = Ad ( U ) ( F ) and ρ ′ = Ad ( U ) ◦ ρ [HR00, Definition 8.2.1]; • operator homotopic if ρ = ρ ′ , H = H ′ , and there exists a norm-continuous path ( F t ) t ∈ [0 , in B ( H ) suchthat F = F , F = F ′ and, for every t ∈ [0 , ρ, H, F t ) is a p -multigraded Fredholm module over A [HR00,Definition 8.2.2].The notion of stable homotopy is defined in terms of unitary equivalence and operator homotopy; see [HR00,Proposition 8.2.12]. Definition 5.14.
Suppose that A is a separable C*-algebra, and p ≥ −
1. The relation B p ( A ) of stable homotopy of p -multigraded Fredholm modules over A is the relation defined by setting x B p ( A ) x ′ if and only if there existsa degenerate p -multigraded Fredholm module x over A such that x ⊕ x and x ′ ⊕ x are unitarily equivalent to apair of operator homotopic p -multigraded Fredholm modules over A .One has that B p ( A ) is an equivalence relation on F p ( A ); see [HR00, Proposition 8.2.12]. Furthermore, B p ( A ) isan analytic equivalence relation, as it follows easily from the definition and the fact that the set D p ( A ) of degenerate p -multigraded Fredholm modules is a Borel subset of F p ( A ), and the set of norm-continuous paths in B ( H ) is aBorel subset of the C*-algebra C β ([0 , , B ( H )) = M ( C ([0 , , K ( H ))) of strictly continuous bounded functions[0 , → B ( H ) by Corollary 2.26.5.3. The Kasparov K -homology groups. We now recall the definition of the Kasparov K-homology groups interms of Fredholm modules; see [HR00, Definition 8.2.5 and Proposition 8.2.12].
Definition 5.15.
Let A be a separable C*-algebra, and fix p ≥ −
1. The
Kasparov K- homology group KK − p ( A ; C )is the semidefinable abelian group obtained as the quotient of the standard Borel space F p ( A ) by the analyticequivalence relation B p ( A ) of stable homotopy of p -multigraded Fredholm modules, where the group operationon KK − p ( A ; C ) is induced by the Borel function F p ( A ) × F p ( A ) → F p ( A ), ( x, x ′ ) x ⊕ x ′ , and the functionKK − p ( A ; C ) → KK − p ( A ; C ) that maps each element to its additive inverse is induced by the Borel functionF p ( A ) → F p ( A ), x x op .The fact that KK − p ( A ; C ) is indeed a group is the content of [HR00, Proposition 8.2.10, Corollary 8.2.11,Proposition 8.2.12]. The trivial element of KK − p ( A ; C ) is given by the B p ( A )-class of any degenerate Fredholmmodule. The assignment A KK − p ( A ; C ) is easily seen to be a contravariant functor from separable C*-algebrasto semidefinable groups. We will later show in Proposition 5.16 that KK − p ( A ; C ) is in fact a definable group.Suppose that A is a separable C*-algebra. Fix a representation ρ A : A → B ( H A ) of A that is the restriction to A of an ample representation of the unitization A + of A . We then let ρ A ⊕ ρ A be the corresponding representation(by even operators) on the graded Hilbert space H A ⊕ H A . Consider also the Paschke dual algebra D ( A ) = D ρ A ( A )associated with ρ A ; see Section 2.10There is a natural definable group homomorphism Φ : K ( D ( A )) → KK ( A, C ), [ P ] [ x P ], defined as follows.Given a projection P in D ( A ), define x P to be the ungraded Fredholm module ( ρ A , H A , P − I ) over A ; see [HR00,Example 8.1.7]. We also have a natural definable group homomorphism Φ : K ( D ( A )) → KK ( A, C ), [ U ] [ x U ],defined as follows. Given a unitary U in D ( A ), define x U to be the graded Fredholm module ( ρ A ⊕ ρ A , H A ⊕ H A , F U )where H A ⊕ H A is graded by I H A ⊕ ( − I H A ), and F U = (cid:20) U ∗ U (cid:21) ;see [HR00, Example 8.1.7]. Then it is shown in [HR00, Theorem 8.4.3] that Φ and Φ are in fact group isomorphism.From this, we obtain the following. Proposition 5.16.
Suppose that A is a separable C*-algebra and i ∈ { , } . Then KK i ( A ; C ) is a definable group,naturally definably isomorphic to K i ( A ) .Proof. Since K − i ( D ( A )) = K i ( A ) is a definable group and Φ i : K − i ( D ( A )) → KK i ( A ; C ) is a definable groupisomorphism, it follows from Corollary 1.14 that KK i ( A ; C ) is a definable group, naturally definably isomorphic toK i ( A ). (cid:3) Remark 5.17.
Suppose that A is a separable C*-algebra, and p ≥ −
1. A p -multigraded Fredholm module ( ρ, H, F )over A is self-adjoint if F is self-adjoint, and contractive if F is contractive [HR00, Definition 8.3.1]. A self-adjoint,contractive p -multigraded Fredholm module ( ρ, H, F ) is involutive if F = 1 [HR00, Definition 8.3.4]. Kasparov’sK-homology groups can be normalized by requiring that the Fredholm modules be involutive. This means that oneobtain the same definable abelian group (up to a natural isomorphism) if one only considers in the definition ofthe Kasparov K-homology groups involutive Fredholm modules, where also stable homotopy is defined in terms ofinvolutive Fredholm modules; see [HR00, Lemma 8.3.5].A graded Fredholm module ( ρ, H, F ) over A is balanced if there is a separable Hilbert space H ′ such that H = H ′ ⊕ H ′ is graded by I H ′ ⊕ ( − I H ′ ), and ρ = ρ + ⊕ ρ − , where ρ + and ρ − are the same representation of A on H ′ . Then one has that KK ( A ; C ) can be normalized by requring that the graded Fredholm modules be involutiveand balanced [HR00, Proposition 8.3.12].Suppose that A is a separable C*-algebra. Fix p ≥
0. If x = ( ρ, H, F ) is a p -multigraded Fredholm module over A , then one can assign to it the ( p + 2)-multigraded Fredholm module x ′ = ( ρ ⊕ ρ op , H ⊕ H op , F ⊕ F op ) where H ⊕ H op is ( p + 2)-multigraded by the operators ε i ⊕ ε op i for 1 ≤ i ≤ p together with (cid:20) I − I (cid:21) and (cid:20) iIiI (cid:21) .When p = − x ′ to be the to be the 1-graded Fredholm module ( ρ ⊕ ρ, H ⊕ H, F ⊕ F ) where H ⊕ H is graded by I H ⊕ ( − I H ) and 1-multigraded by the odd operator (cid:20) iIiI (cid:21) .This gives for p ≥ − − p ( A ; C ) → KK − p − ( A ; C ) [HR00, Proposition8.2.13]. From this, Proposition 1.14, and Proposition 5.16, we obtain that, for p ≥ −
1, KK − p ( A ; C ) is a definablegroup, naturally definably isomorphic to K − p − ( A ).5.4. Relative Kasparov K -homology. The relative
Kasparov K-homology groups are defined as above, by re-placing Fredholm modules with relative
Fredholm modules; see [HR00, Definition 8.5.1].
Definition 5.18.
Suppose that (
A, J ) is a separable C*-pair. An ungraded Fredholm module over (
A, J ) is a triple( ρ, H, F ) where: • H is a separable Hilbert space H d for some d ∈ ω ∪ {ℵ } ; • ρ : A → B ( H ) is a *-homomorphism; • F ∈ B ( H ) satisfies (cid:0) F − (cid:1) ρ ( j ) ≡ ( F − F ∗ ) ρ ( j ) ≡ [ F, ρ ( a )] ≡ K ( H ) for every a ∈ A and j ∈ J .A graded Fredholm module over ( A, J ) is a triple ( ρ, H, F ) where: • H is a separable graded Hilbert space of the form ( H d , γ ) for some d ∈ ω ∪ {ℵ } and some grading operator γ on H d ; • ρ : A → B ( H ) is a *-homomorphism such that, for every a ∈ A , ρ ( a ) ∈ B ( H ) + is even ; • F ∈ B ( H ) is an odd operator that satisfies (cid:0) F − (cid:1) ρ ( j ) ≡ ( F − F ∗ ) ρ ( j ) ≡ [ F, ρ ( a )] ≡ K ( H ) forevery a ∈ A and j ∈ J .As above, one can consider the definable group KK − ( A, J ; C ) whose elements are stable homotopy equivalenceclasses of Fredholm modules over ( A, J ). Considering graded
Fredholm modules over (
A, J ) one obtains the definablegroup KK ( A, J ; C ). These groups are called the relative Kasparov K-homology groups of the pair ( A, J ), and turnout to be naturally definably isomorphic to the relative K-homology groups K ( A, J ) and K ( A, J ); see [HR00,Section 8.5]. More generally, one can define KK − p ( A, J ; C ) in terms of p -multigraded Fredholm modules over ( A, J ).In the Kasparov picture, the excision isomorphism KK − p ( A, J ; C ) → K − p ( J ; C ) is induced by the inclusion mapfrom the set of p -multigraded Fredholm modules over ( A, J ) into the set of p -multigraded Fredholm modules over J . (Notice that a Fredholm module over ( A, J ) is, in particular, a Fredholm module over J .) EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 43 KK h -cycles and K -homology. Suppose that A is a separable C*-algebra. Let H be the separable infinite-dimensional Hilbert space. A KK h -cycle for A is a pair ( φ + , φ − ) of *-homomorphisms A → B ( H ) such that φ + ( a ) ≡ φ − ( a ) mod K ( H ) for every a ∈ A ; see [JT91, Definition 4.1.1]. Define F ( A ; C ) to be the standard Borelspace of KK h -cycles for A . The standard Borel structure on F ( A ; C ) is induced by the Polish topology obtained bysetting ( φ ( i )+ , φ ( i ) − ) → ( φ + , φ − ) if and only if, for every a ∈ A , φ ( i )+ ( a ) → φ + ( a ) and φ ( i ) − → φ − ( a ) in the strong-*topology, and ( φ ( i )+ − φ ( i ) − ) ( a ) → ( φ + − φ − ) ( a ) in norm. We regard F ( A ; C ) as a Polish space with respect to sucha topology.Define ∼ to be the relation of homotopy for elements of the Polish space F ( A ; C ). Thus, for x, x ′ ∈ F ( A ; C ), x ∼ x ′ if and only if there exists a continuous path ( x t ) t ∈ [0 , in F ( A ; C ) such that x = x and x = x ′ ; see[JT91, Definition 4.1.2]. As discussed in Section 2.3 one can regard a strong-* continuous path ( φ t ) t ∈ [0 , of *-homomorphisms A → B ( H ) as an element of the unit ball of C β ([0 , , B ( H )) = M ( C ([0 , , K ( H ))). This allowsone to regard the set of such paths as a Polish space endowed with the strict topology on Ball ( C β ([0 , , B ( H ))),such that norm-continuous paths form a Borel subset by Corollary 2.26. It can be deduced from these observationsthat the relation ∼ of homotopy in F ( A ; C ) is an analytic equivalence relation.One lets KK h ( A ; C ) be the semidefinable set obtained as a quotient of the Polish space F ( A ; C ) by the an-alytic equivalence relation ∼ [JT91, Definition 4.1.3]. One has that KK h ( A ; C ) is a semidefinable group, wherethe group operation is induced by the Borel function F ( A ; C ) × F ( A ; C ) → F ( A ; C ), (( φ + , φ − ) , ( ψ + , ψ − )) (Ad ( V ) ◦ ( φ + ⊕ ψ + ) , Ad ( V ) ◦ ( φ − ⊕ ψ − )), where V : H ⊕ H → H is a fixed surjective linear isometry, and thefunction mapping each element to its additive inverse is induced by the Borel function KK h ( A ; C ) → KK h ( A ; C ),( φ + , φ − ) ( φ − , φ + ); see [JT91, Proposition 4.1.5]. The trivial element of KK h ( A ; C ) is the homotopy class of(0 , A KK h ( A ; C ) gives a contravariant functor from separable C*-algebras to semidefinablegroups.Let A be a separable C*-algebra. We now observe that KK h ( A ; C ) is in fact a definable group, definablyisomorphic to KK ( A ; C ) and hence to K ( A ). There is a natural definable isomorphism Ψ : KK h ( A ; C ) → KK ( A ; C ) defined as follows; see [JT91, Theorem 4.1.8]. Suppose that ( φ + , φ − ) ∈ F ( A ; C ). Then one can considerthe graded Kasparov module over A defined as ( φ + ⊕ φ − , H ⊕ H, F ) where H ⊕ H is graded by I H ⊕ ( − I H ) and F = (cid:20) I H I H (cid:21) .Then one sets Ψ ([ φ + , φ − ]) = [ φ + ⊕ φ − , H ⊕ H, F ].We now observe that the inverse function Ψ − : KK ( A ; C ) → KK h ( A ; C ) is also definable, as it follows fromthe proof of [JT91, Theorem 4.1.8]. Let ( ρ , H , F ) be a graded Kasparov module, which can be assumed tobe involutive and balanced by normalization and where we can assume H to be infinite-dimensional; see [HR00,Proposition 8.3.12]. Then we have that H = H ⊕ H is graded by I H ⊕ ( − I H ) and ρ +0 = ρ − are the samerepresentation of A on H , and F = (cid:20) u ∗ u (cid:21) for some unitary u ∈ B ( H ′ ). Then by [JT91, E 2.1.3], the Kasparov modules( ρ , H , F ) and (cid:0)(cid:0) Ad( u ) ◦ ρ + (cid:1) ⊕ ρ − , H ⊕ H, F (cid:1) represent the same element of KK ( A ; C ), where as above F = (cid:20) I H I H (cid:21) .One has that Ψ − [ ρ , H , F ] = [((Ad( u ) ◦ ρ + ) , ρ − )]. As the assignment ( ρ , H , F ) ((Ad( u ) ◦ ρ + ) , ρ − ) is givenby a Borel function, this shows that the inverse Ψ − : KK ( A ; C ) → KK h ( A ; C ) is definable. We thus obtain thefollowing. Proposition 5.19.
Let A be a separable C*-algebra. Then KK h ( A ; C ) is a definable group, naturally definablyisomorphic to K ( A ) . Proof.
By the above discussion, the natural definable homomorphism KK h ( A ; C ) → KK ( A ; C ) is an isomorphismin the category of semidefinable groups. Therefore, KK h ( A ; C ) is also a definable group, naturally isomorphic toKK ( A ; C ). As in turn KK ( A ; C ) is naturally definably isomorphic to K ( A ), the conclusion follows. (cid:3) Cuntz’s K -homology. Suppose that
A, B are separable C*-algebras. Let Hom (
A, B ) be the set of *-homomorphisms A → B . Then Hom ( A, B ) is a Polish space when endowed with the topology of pointwise norm-convergence. Two *-homomorphisms φ, φ ′ : A → B are homotopic, in which case we write φ ∼ φ ′ , if they belongto the same path-connected component of Hom ( A, B ). Thus, two *-homomorphism φ, φ ′ : A → B satisfy φ ∼ φ ′ if and only if there exists a continuous path ( λ t ) t ∈ [0 , in Hom( A, B ) such that λ = φ and λ = φ ′ ; see [JT91,Definition 1.3.10]. Such a path ( λ t ) t ∈ [0 , can be thought of as a *-homomorphism λ : A → C ([0 , , B ), where C ([0 , , B ) ∼ = C ([0 , ⊗ B is the C*-algebra of continuous functions [0 , → B . This shows that the relation ∼ of homotopy in Hom ( A, B ) is an analytic equivalence relation. We let [
A, B ] be the semidefinable set of homotopyclasses of *-homomorphisms A → B .Recall that a separable C*-algebra B is stable if B ⊗ K ( H ) is *-isomorphic to B . Suppose in the followingthat B is stable. Thus we have that M ( B ) ⊗ M ( K ( H )) ⊆ M ( B ⊗ K ( H )) ∼ = M ( B ). This implies that one canchoose isometries w , w ∈ M ( B ) satisfying w w ∗ + w w ∗ = 1 and w ∗ i w j = 0 for i, j ∈ { , } distinct. (This isequivalent to the assertion that w , w generate inside M ( B ) a copy of the Cuntz algebra O .) One can then definea *-isomorphism θ : M ( B ) → B , x w xw ∗ + w xw ∗ . A *-isomorphism of this form is called inner ; see [JT91,Definition 1.3.8]. Any two inner *-isomorphisms θ, θ ′ : M ( B ) → B are unitary equivalent, namely there exists aunitary u ∈ M ( B ) such that Ad( u ) ◦ θ = θ ′ [JT91, Lemma 1.3.9].Under the assumption that B is stable, one can endow the semidefinable set [ A, B ] with the structure ofsemidefinable semigroup. The operation on [
A, B ] is induced by the Borel function Hom (
A, B ) → Hom (
A, B ),( φ, ψ ) θ ◦ ( φ ⊕ ψ ), where θ is a fixed inner *-isomorphism M ( B ) → B ; see [JT91, Lemma 1.3.12]. The trivialelement in [ A, B ] is the homotopy class of the zero *-homomorphism. Furthermore, the argument of [JT91, E 4.1.4]shows that [
A, B ] is isomorphic to [ K ( H ) ⊗ A, B ] in the category of semidefinable semigroups.Suppose that A is a separable C*-algebra. Define QA to be the separable C*-algebra A ∗ A , where A ∗ A denotesthe free product of A with itself. We let i, i be the two canonical inclusions of A inside QA . Let qA be the closedtwo-sided ideal of QA generated by the elements of the form i ( a ) − i ( a ) for a ∈ A ; see [JT91, Definition 5.1.1].If B is a separable C*-algebra, and φ, ψ : A → B are *-homomorphism, then there is a unique *-homomorphism Q ( φ, ψ ) : QA → B such that Q ( φ, ψ ) ◦ i = φ and Q ( φ, ψ ) ◦ i = ψ . The restriction of Q ( φ, ψ ) to qA is denoted by q ( φ, ψ ). One has that the range of q ( φ, ψ ) is contained in an ideal J of B if and only if φ ( a ) ≡ ψ ( a ) mod J forevery a ∈ A , in which case q ( φ, ψ ) ∈ Hom ( qA, J ). One has that qA is the kernel of the map Q (id A , id A ) : QA → A ;see [JT91, Lemma 5.1.2].If B is a separable C*-algebra, then the semidefinable semigroup [ qA, K ( H ) ⊗ B ] is in fact a semidefinable group,where the function that maps each element to its additive inverse is induced by the Borel functionHom ( qA, K ( H ) ⊗ B ) → Hom ( qA, K ( H ) ⊗ B ) , φ
7→ − φ ;[JT91, Theorem 5.1.6]. The proof of [JT91, Theorem 5.1.12] shows that [ qA, K ( H ) ⊗ B ] is isomorphic in the cate-gory of semidefinable groups to [ qA, K ( H ) ⊗ qB ]. In turn, [ qA, K ( H ) ⊗ qB ] is isomorphic to [ K ( H ) ⊗ qA, K ( H ) ⊗ qB ]in the category of semidefinable groups by [JT91, E 4.1.4].Observe that, for a fixed C*-algebra B , the assignment A [ qA, K ( H ) ⊗ B ] is a contravariant functor fromC*-algebras to semidefinable groups. Suppose that A is a separable C*-algebra. Then there is a natural definablehomomorphism S : KK h ( A ; C ) → [ qA, K ( H )] defined by setting S ([ φ + , φ − ]) = [ ψ ] where ψ = q ( φ + , φ − ) ∈ Hom ( qA, K ( H )). One has that in fact S is a group isomorphism [JT91, Theorem 5.2.4]. Therefore, we obtain fromProposition 5.19 and Corollary 1.14 the following. Proposition 5.20.
Suppose that A is a separable C*-algebra. Then [ qA, K ( H )] is a definable group, naturallydefinably isomorphic to K ( A ) . This description of K-homology is called Cuntz’s picture, as it was introduced by Cuntz in [Cun87]; see also[Bla86, Section 17.6] and [Cun83, Cun84, Zek89]. Using the Cuntz picture, one can easily define the more generalKasparov KK-groups KK ( A, B ) for separable C*-algebras
A, B , by settingKK ( A ; B ) = [ qA, K ( H ) ⊗ B ] ∼ = [ K ( H ) ⊗ qA, K ( H ) ⊗ qB ] . EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 45
These are semidefinable groups, although we do not know whether they are definable groups when B is an arbitraryseparable C*-algebra. In particular, one has that KK ( A ; C ) ∼ = K ( A ) and KK ( A, C ( R )) ∼ = KK ( SA ; C ) ∼ =K ( A ). The K-theory groups are also recovered as particular instances of the KK-groups, as KK ( C ; A ) ∼ = K ( A )and KK ( C ( R ) ; A ) ∼ = KK ( C ; SA ) ∼ = K ( A ).Given separable C*-algebras A, B, C , composition of *-homomorphisms K ( H ) ⊗ qA → K ( H ) ⊗ qB and K ( H ) ⊗ qB → K ( H ) ⊗ qC induces a definable bilinear pairing (Kasparov product)KK ( A ; B ) × KK ( B ; C ) → KK ( A ; C ) .In particular, KK ( A ; A ) is a (semidefinable) ring, with identity element A corresponding to the identity mapof K ( H ) ⊗ qA . The KK-category of C*-algebras is the category enriched over the category of (semidefinable)abelian groups that has separable C*-algebras as objects and KK-groups as hom-sets. Two separable C*-algebrasare KK-equivalent if they are isomorphic in the KK-category of C*-algebras.By way of the Kasparov product and the natural isomorphisms K ( A ) ∼ = KK ( A ; C ) and K ( A ) ∼ = KK ( SA ; C ),one can regard K-homology as a contravariant functor from the KK-category of separable C*-algebras to thecategory of definable groups, and K-theory as a covariant functor from the KK-category of separable C*-algebras tothe category of countable groups. In particular, KK-equivalent C*-algebras have definably isomorphic K-homologygroups, and isomorphic K-theory groups.6. Properties of definable K -homology In this section we consider several properties of definable K-homology, which can be seen as definable versionsof the properties of an abstract cohomology theory in the sense of [Sch84] that is C*-stable in the sense of [Cun87].6.1.
Products.
Suppose that ( X i ) i ∈ ω is a sequence of semidefinable sets X i = ˆ X i /E i . Then the product Q i ∈ ω X i is the semidefinable set ˆ X/E where ˆ X = Q i ∈ ω ˆ X i and E is the (analytic) equivalence relation on ˆ X defined bysetting ( x i ) E ( y i ) if and only if ∀ i ∈ ω , x i E i y i . If, for every i ∈ ω , G i is a semidefinable group, then Q i ∈ ω G i is asemidefinable group when endowed with the product group operation.Suppose that ( A i ) i ∈ ω is a sequence of separable C*-algebra. Define the direct sum L i ∈ ω A i to be the C*-algebra A consisting of the sequences ( a i ) i ∈ ω ∈ Q i ∈ ω A i such that k a i k →
0; see [HR00, Definition 7.4.1]. If B is a separableC*-algebra, then the canonical maps A i → A induce an isomorphism of Polish spacesHom ( A, B ) → Y i ∈ ω Hom ( A i , B ) .When A i is commutative with spectrum X i , then A is commutative with spectrum the disjoint union of X i for i ∈ ω . The following result can be seen as a noncommutative version of the Cluster Axiom for a homology theoryfor pointed compact spaces from [Mil95]. Proposition 6.1.
Suppose that ( A i ) i ∈ ω is a sequence of separable C*-algebras, and set A = L i ∈ ω A i . Fix p ∈ { , } .Then Q i ∈ ω K p ( A i ) is a definable group. Furthermore the canonical maps A i → A for i ∈ ω induce a natural definableisomorphism K p ( A ) → Y i ∈ ω K p ( A i ) .Proof. Since K p ( A ) is a definable group, it suffices to prove the second assertion. After replacing A with itssuspension, it suffices to consider the case when p = 0. In this case, we can replace K with KK h by Proposition5.19. Recall that we let F ( A ; C ) be the space of KK h -cycles for A . The canonical maps A i → A induce anisomorphism of Polish spaces F ( A ; C ) → Y i ∈ ω F ( A i ; C ) .In turn, this induces a definable isomorphism of the spaces of homotopy classes.KK h ( A ) → Y i ∈ ω KK h ( A i ) .This concludes the proof. (cid:3) Homotopy-invariance.
Suppose that
A, B are separable C*-algebra. Recall that Hom (
A, B ) is a Polishspace when endowed with the topology of pairwise convergence. Thus, α, β ∈ Hom (
A, B ) if there exists a path inHom (
A, B ) from α to β ; see [HR00, Definition 4.4.1]. This can be thought of as an element γ of Hom ( A, IB ) suchthat ev ◦ γ = α and ev ◦ γ = β where IB = C ([0 , , B ) and ev t : IB → B , f f ( t ) for t ∈ [0 , A, B ]be the semidefinable set of homotopy classes of *-homomorphisms A → B . The homotopy category of C*-algebrashas separable C*-algebras as objects and homotopy classes of *-homomorphisms as morphisms. Two C*-algebrasare homotopy equivalent if they are isomorphic in the homotopy category of C*-algebras [HR00, Definition 4.4.7]. Proposition 6.2.
For p ∈ { , } , the K -homology functor K p ( − ) from separable C*-algebras is homotopy-invariant.Proof. As in the case of the proof of Proposition 6.1, it suffices to show that the functor KK h ( − ) is homotopyinvariant, which is an immediate consequence of the definition. (cid:3) Suppose that B is a separable C*-algebra. Recall that the suspension SB of B can be seen as the C*-subalgebraof IB consisting of f ∈ IB such that f (0) = f (1) = 0. Then [ A, SB ] is a semidefinable abelian group, where thegroup operation is induced by the Borel function ( f, g ) m ( f, g ) where m ( f, g ) ( t ) = (cid:26) f (2 t ) t ∈ [0 , /
2] , g (2 t − t ∈ [1 / ,
1] .The function that assigns each element to its additive inverse is induced by the Borel function f b f where b f ( t ) = f (1 − t ) .The trivial element of [ A, SB ] is the homotopy class of 0. For p ∈ { , } , there map [ A, SB ] → K p ( SB, A ) is agroup homomorphism [Sch84, Proposition 6.3].A separable C*-algebra A is contractible if it is homotopy equivalent to the zero C*-algebra; see [HR00, Definition4.4.4]. By homotopy invariance, K p ( A ) = { } whenever A is contractible and p ∈ { , } . In particular, if ( A, J ) isa separable semi-split C*-pair such that A is contractible, the boundary homomorphism K p ( J ) → K p ( A/J ) is adefinable isomorphism.If A is a separable, nuclear C*-algebra, then its cone CA is the C*-subalgebra of IA consisting of f ∈ IA suchthat f (1) = 0. This is a contractible C*-algebra [HR00, Example 4.4.6], and0 → SA → CA → A → CA → A is the map ev . The boundary homomorphism σ A : K p ( SA ) → K p +1 ( A ) isthus an isomorphism; see [Sch84, Theorem 6.5].6.3. Mapping cones.
Suppose that
A, B are separable, nuclear C*-algebras, and f : A → B is a *-homomorphism.The mapping cone Cf = { ( x, y ) ∈ CB ⊕ A : f ( y ) = ev ( x ) } of f is obtained as the pullback of ev : CB → B and f : A → B . As such, it is endowed with canonical*-homomorphisms Cf → CB and Cf → A ; see [Sch84, Definition 2.1]. We have a natural exact sequence0 → SB → Cf → A → SB → Cf , x ( x, p ( SB ) → K p +1 ( A ).Considering the commutative diagram0 → SB → Cf → A → ↓ ↓ ↓ f → SB → CB → B → SB → SB is the identity map, and Cf → CB , ( x, y ) x , we obtain by naturality of the six-term exactsequence in K-homology that the boundary morphism K p ( SB ) → K p +1 ( A ) is equal to the compositionK p +1 ( f ) ◦ σ B : K p ( SB ) → K p +1 ( B ) → K p +1 ( A ) .The same argument together with the Five Lemma [Rot09, Proposition 2.72] shows that K p ( f ) is an isomorphismfor every p ∈ { , } if and only if K p ( Cf ) = { } for every p ∈ { , } ; see [Sch84, Theorem 6.5]. EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 47 If f : A → B is a surjective *-homomorphism with kernel J , then considering the exact sequence0 → J → Cf → CB → J → Cf induces an isomorphism K p ( J ) → K p ( Cf ). Similarly, if J is an ideal of A and f : J → A is the inclusion, then considering the exact sequence0 → CJ → Cf → S ( A/J ) → Cf → S ( A/J ) induces an isomorphism K p ( Cf ) → K p ( S ( A/J )); see [Sch84, Proposition 6.6].6.4.
Long exact sequence of a triple.
Consider a triple J ⊆ H ⊆ A where A is a separable, nuclear C*-algebraand J and H are closed two-sided ideals of A . Then we have a commutative diagram0 → H → A → A/H → ↓ ↓ ↓ → H/J → A/J → A/H → p ( H/J ) → K − p ( A/H )is equal to the composition of the map K p ( H/J ) → K p ( H ) induced by the quotient map with the boundary mapK p ( H ) → K − p ( A/H ); see [Sch84, Theorem 6.10].6.5.
Mayer–Vietoris sequence.
Consider separable, nuclear C*-algebras
P, A , A , B , and *-homomorphisms f i : A i → B and g i : P → A i for i ∈ { , } . Suppose that f , f are surjective, and P A A B g g f f is a pushout diagram. Then there is a six-term exact sequence of definable group homomorphismsK ( B ) K ( A ) ⊕ K ( A ) K ( P )K ( P ) K ( A ) ⊕ K ( A ) K ( B ) ∂ ∂ see [Sch84, Theorem 6.11]. The definable group homomorphism K p ( B ) → K p ( A ) ⊕ K p ( A ) is ( − K p ( f ) , K p ( f )),the definable group homomorphism K p ( A ) ⊕ K p ( A ) → K p ( P ) is K p ( g ) + K p ( g ). Furthermore, the definablegroup homomorphism ∂ p : K p ( P ) → K − p ( B ) is defined as follows. Let g : P → A ⊕ A be defined by x ( g ( x ) , g ( x )). Consider the corresponding mapping cone Cg . We can regard Cg as the set of triples ( ξ , ξ , x ) ∈ CA ⊕ CA ⊕ P such that ( ξ (0) , ξ (0)) = g ( x ). We have a *-homomorphism ψ : Cg → SB defined by setting ψ ( ξ , ξ , x ) ( t ) = (cid:26) f ( ξ (1 − t )) , t ∈ [0 , /
2] ; f ( ξ (2 t − t ∈ [1 / ,
1] .Then we have a natural short exact sequence0 → CJ ⊕ CJ → Cg ψ → SB → J i = Ker ( f i ) for i ∈ { , } ; see [Sch84, Proposition 4.5]. Thus, ψ induces a definable isomorphism K p ( SB ) → K p ( Cg ). The definable group homomorphism ∂ p : K p ( P ) → K − p ( B ) is defined as the composition of definablehomomorphisms K p ( P ) → K p ( Cg ) → K p ( SB ) → K − p ( B )where the map K p ( P ) → K p ( Cg ) is associated with the canonical *-homomorphism Cg → P as in the definitionof mapping cone, the map K p ( Cg ) → K p ( SB ) is the inverse of the definable isomorphism K p ( SB ) → K p ( Cg )induced by ψ , and the map σ B : K p ( SB ) → K − p ( B ) is the suspension isomorphism; see the proof of [Sch84,Theorem 6.11]. The Milnor sequence of an inductive sequence.
A tower of countable abelian groups is a sequence A = (cid:0) A ( n ) , p ( n,n +1) (cid:1) of countable abelian groups and group homomorphism p ( n,n +1) : A ( n +1) → A ( n ) . Given sucha tower we let p ( n,n ) be the identity map of A ( n ) and, for n < m , p ( n,m ) be the composition p ( n,n +1) ◦ · · · ◦ p ( m − ,m ) .Towers of countable groups form a category. A morphism from A = (cid:0) A ( n ) , p ( n,n +1) (cid:1) to B = (cid:0) B ( k ) , p ( k,k +1) (cid:1) isrepresented by a sequence (cid:0) n k , f ( k ) (cid:1) k ∈ ω where ( n k ) is an increasing sequence in ω and f ( k ) : A ( n k ) → B ( k ) is agroup homomorphism. Two such sequences (cid:0) n k , f ( k ) (cid:1) k ∈ ω and (cid:0) n ′ k , f ′ ( k ) (cid:1) k ∈ ω represent the same morphism if thereexists an increasing sequence ( n ′′ k ) k ∈ ω in ω such that n ′′ k ≥ max { n k , n ′ k } and f ( k ) p ( n k ,n ′′ k ) = f ( k ) p ( n ′ k ,n ′′ k ) for every k ∈ ω . The identity morphism and composition of morphisms are defined in the obvious way.Given a tower A of countable abelian groups, one lets lim A be the definable group, which is in fact a groupwith Polish cover (see Remark 1.17), defined as follows. Consider Z ( A ) to be the product group Y n ∈ ω A ( n ) endowed with the product topology, where each A ( n ) is endowed with the discrete topology. Define B ( A ) to bethe Polishable Borel subgroup of Z ( A ) obtained as an image of the continuous group homomorphismΦ A : Y n ∈ ω A ( n ) → Z ( A ) , ( x n ) (cid:16) x n − p ( n,n +1) ( x n +1 ) (cid:17) n ∈ ω .Then lim A is the corresponding definable group Z ( A ) / B ( A ). The assignment A lim A is easily seen tobe a functor from the category of towers of countable abelian groups to the category of definable groups; see also[BLP20, Section 5].Given a tower A of countable abelian groups, we can also consider the (inverse) limit lim A . This is the Polishabelian group obtained as the kernel of the continuous group homomorphism Φ A described above. The assignment A lim A is a functor from the category of towers of countable abelian groups to the category of Polish abeliangroups.Suppose that ( A n , ϕ n ) n ∈ ω is an inductive sequence of separable, nuclear C*-algebras, and let A = colim n ( A n , ϕ n )be its inductive limit. If K p ( A n ) is countable for every n ∈ ω , then (K p ( A n )) n ∈ ω is a tower of countable abeliangroups, where p ( n,n +1) : K p ( A n +1 ) → K p ( A n ) is induced by ϕ n : A n → A n +1 . The assignment ( A n , ϕ n ) n ∈ ω (K p ( A n )) n ∈ ω defines a functor from the category of inductive sequences of separable C*-algebras with countableK-homology groups to the category of towers of countable abelian groups. The Milnor sequence for ( A n , ϕ n ) n ∈ ω describes K p ( A ) as an extension of groups defined in terms of (K p ( A n )) n ∈ ω ; see [Sch84, Theorem 7.1]. The proofis inspired by Milnor’s argument for the corresponding result about Steenrod homology [Mil95]; see also [Mil62].We let N denote the set of natural numbers not including zero, and ω = N ∪ { } . Proposition 6.3.
Suppose that ( A n , ϕ n ) n ∈ N is an inductive sequence of separable, nuclear C*-algebras with count-able K -homology groups, and A is the inductive limit of ( A n , ϕ n ) n ∈ ω . Then for p ∈ { , } there is a natural shortexact sequence of definable group homomorphisms → lim n K − p ( A n ) → K p ( A ) → lim n K p ( A n ) → where the homomorphism K p ( A ) → lim n K p ( A n ) is induced by the canonical maps A n → A . The assertion that the group homomorphisms in Proposition 6.3 are definable is a consequence of the proofof [Sch84, Theorem 7.1]. This involves the notion of mapping telescope T ( A ) of an inductive sequence of A =( A n , ϕ n ) n ∈ N of separable C*-algebras; see [Sch84, Definition 5.2]. Without loss of generality, we can assume that A = { } . Let A be the corresponding direct limit and ϕ ( ∞ ,n ) : A n → A be the canonical maps. For n 1) with t = 0 converging to 1. Let Q n ∈ ω C ([ t n , t n +1 ] , A n +1 ) be theproduct of ( C ([ t n , t n +1 ] , A n +1 )) n ∈ ω in the category of C*-algebras. Define then ˜ T ( A ) to be the C*-subalgebraof Q n ∈ ω C ([ t n , t n +1 ] , A n +1 ) consisting of those elements ( ξ n ) n ∈ ω such that, for every n ∈ ω , ϕ n +1 ( ξ n ( t n +1 )) = ξ n +1 ( t n +1 ). An element ( ξ n ) n ∈ ω of ˜ T ( A ) can be seen as a function ξ : [0 , → S n ∈ ω A n +1 where, for n ∈ ω and t ∈ [ t n , t n +1 ) one sets ξ ( t ) := ξ n ( t ). The function ξ ∞ : [0 , → A defined by ξ ∞ ( t ) = ϕ ( ∞ ,n +1) ( ξ ( t )) for t ∈ [ t n , t n +1 ) is then continuous. EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 49 The mapping telescope T ( A ) consists of the set of pairs ( ξ, a ) ∈ ˜ T ( A ) ⊕ A such that:(1) for every ε > n ∈ ω such that, for n ≥ m ≥ n and for t ∈ [ t n , t n +1 ] and s ∈ [ t m , t m +1 ], (cid:13)(cid:13) ϕ ( n +1 ,m +1) ( ξ m ( s )) − ξ n ( t ) (cid:13)(cid:13) < ε ,and(2) lim t → ξ ∞ ( t ) = a .Then one has that T ( A ) is a contractible separable C*-algebra; see [Sch84, Lemma 5.4]. Define the surjective*-homomorphism e : T ( A ) → A , ( ξ, a ) a , and set J = Ker ( e ) ⊆ T ( A ). We also have a map p : J → L n ∈ ω A n +1 , ξ ( ξ n ( t n +1 )) n ∈ ω ; see [Sch84, Lemma 5.5]. As T ( A ) is contractible, the short exact sequence0 → J → T ( A ) → A → ∂ : K − p ( J ) → K p ( A ).For n ∈ ω define M n ⊆ C ([ t n , t n +1 ] , A n +1 ) ⊕ A n to be the C*-subalgebra consisting of ( ξ, a ) such that ξ ( t n ) = ϕ n ( a ). The *-homomorphism M n → A n , ( ξ, a ) a is a homotopy equivalence with homotopy inverse A n → M n , a ( ξ, a ) where ξ ( t ) = ϕ n ( a ) for t ∈ [0 , A n → M n → A n is the identity,while the composition M n → A n → M n maps ( ξ, a ) to ( ξ ′ , a ) where ξ ′ ( t ) = ξ ( t n ) = ϕ n ( a ) for t ∈ [0 , φ t ) s ∈ [0 , defined by φ s ( ξ, a ) = ( ξ s , a ) where ξ s ( t n + t ( t n +1 − t n )) = ξ ( t n + st ( t n +1 − t n ))for s, t ∈ [0 , D := M n ∈ ω M n +1 D := M n ∈ ω M n . B := M n ∈ ω A n .As in [Sch84, Lemma 5.7], we have a pullback diagram J D D B g g f f where: • g : J → D is defined by ( ξ k ) k ∈ ω ( η n , b n ) n ∈ ω where ξ k ∈ C ([ t k , t k +1 ] , A k +1 ) for k ∈ ω and ( η n , b n ) = ( ξ n +1 , ξ n ( t n +1 )) ∈ M n +1 for n ∈ ω ; • g : J → D is defined by ( ξ k ) k ∈ ω ( η n , b n ) n ∈ ω where ξ k ∈ C ([ t k , t k +1 ] , A k +1 ) for k ∈ ω , and ( η n , b n ) = ( ξ n , ξ n − ( t n )) ∈ M n ; • f : D → B is defined by ( η n , b n ) n ∈ ω ( c n ) n ∈ ω where ( η n , b n ) ∈ M n +1 , c = 0, c n +1 = b n , and c n +2 = η n ( t n +2 ) for n ∈ ω ; • f : D → B is defined by ( η n , b n ) n ∈ ω ( c n ) n ∈ ω where ( η n , b n ) ∈ M n , c n = b n , and c n +1 = η n ( t n +1 ) for n ∈ ω .We thus have a corresponding Mayer–Vietoris definable six-term exact sequenceK ( B ) K ( D ) ⊕ K ( D ) K ( J )K ( J ) K ( D ) ⊕ K ( D ) K ( B ) ∂ ∂ associated with it. Combining this with the definable isomorphism K − p ( J ) → K p ( A ) as above, and with thedefinable isomorphisms K p ( B ) ∼ = Y n ∈ ω K p ( A n )K p ( D ) ⊕ K p ( D ) ∼ = Y n ∈ ω K p ( A n ) ⊕ Y n ∈ ω K p ( A n +1 ) ∼ = Y n ∈ ω K p ( A n )obtained from Proposition 6.1 and from the homotopy equivalences M n → A n for n ∈ ω , one obtains a definablesix-term exact sequence Q n ∈ ω K ( A n ) Φ → Q n ∈ ω K ( A n ) → K ( A ) ↑ ↓ ∂ K ( A ) ← Q n ∈ ω K ( A n ) Φ ← Q n ∈ ω K ( A n ) .As in the proof of [Sch84, Theorem 7.1], the group homomorphismΦ p : Y n ∈ ω K p ( A n ) → Y n ∈ ω K p ( A n )for p ∈ { , } is given by ( x n ) ( x n − K p ( ϕ n ) ( x n +1 )) n ∈ ω whereas the boundary homomorphism K ( A ) → Q n ∈ ω K ( A n ) is induced by the canonical maps A n → A . Thus,by definition of lim and lim of the tower (K p ( A n )) n ∈ ω we have that Φ and Φ yield a definable exact sequence0 → lim n K − p ( A n ) → K p ( A ) → lim n K p ( A n ) → C*-stability. Suppose that A is a separable C*-algebra, and H is a (not necessarily infinite-dimensional)separable Hilbert space. If e ∈ K ( H ) is a rank one projection, then we can define a *-homomorphism e A : A → K ( H ) ⊗ A , a e ⊗ a . In turn, this induces a definable homomorphism K p ( K ( H ) ⊗ A ) → K p ( A ). The stability —or C*-stability [Cun87]—property of K-homology asserts that such a definable homomorphism K p ( K ( H ) ⊗ A ) → K p ( A ) is in fact an definable isomorphism; see [HR00, Theorem 9.4.1]. Proposition 6.4. Suppose that A is a separable C*-algebra, H is a separable Hilbert space, e ∈ K ( H ) is a rankone projection, and e A : A → K ( H ) ⊗ A is the *-homomorphism defined by a e ⊗ a . Then the induced map K p ( e A ) : K p ( K ( H ) ⊗ A ) → K p ( A ) is a definable isomorphism.Proof. It is easy to see that one can reduce to the case when H is infinite-dimensional. After replacing A with itsstabilization, we can assume that p = 0. As K ( − ) is naturally isomorphic to KK h ( − ; C ), it suffices to prove thecorresponding statement for KK h ( − ; C ). One can then proceed as in [JT91, E 4.1.3]. Fix an infinite-dimensionalseparable Hilbert space H , and let KK h ( − ; C ) be defined with respect to H . Consider the canonical inclusions K ( H ) ⊗ B ( H ) ⊆ B ( H ) ⊗ B ( H ) ⊆ B ( H ⊗ H ) and the injective *-homomorphism e K ( H ) : K ( H ) → K ( H ) ⊗ K ( H ), x e ⊗ x . Consider also the *-isomorphism λ : K ( H ) → K ( H ) ⊗ K ( H ) ∼ = K ( H ⊗ H ) defined by setting λ = Ad ( V ) ◦ e K ( H ) where V ∈ B ( H ⊗ H ) is an isometry with V V ∗ = e ⊗ I . Then λ extends to strict *-isomorphism ¯ λ = Ad ( V ) ◦ ¯ e K ( H ) : B ( H ) → B ( H ⊗ H ), where ¯ e K ( H ) : B ( H ) → B ( H ⊗ H ) is the strict extensionof e K ( H ) : K ( H ) → K ( H ) ⊗ K ( H ) ∼ = K ( H ⊗ H ).One can then consider the definable homomorphism G : KK h ( A ; C ) → KK h ( K ( H ) ⊗ A ; C ) induced by the Borelfunction F ( A ; C ) → F ( K ( H ) ⊗ A ; C ) , ( φ + , φ − ) (¯ λ − ◦ (cid:0) id K ( H ) ⊗ φ + (cid:1) , ¯ λ − ◦ (cid:0) id K ( H ) ⊗ φ − (cid:1) ).Then we have that KK h ( e A ; C ) ◦ G : KK h ( A ; C ) → KK h ( A ; C ) is equal to the identity map. Indeed, KK h ( e A ; C ) ◦ G is induced by the function F ( A ; C ) → F ( A ; C ) , ( φ + , φ − ) (¯ λ − ◦ (cid:0) id K ( H ) ⊗ φ + (cid:1) ◦ e A , ¯ λ − ◦ (cid:0) id K ( H ) ⊗ φ − (cid:1) ◦ e A ).We have that(¯ λ − ◦ (cid:0) id K ( H ) ⊗ φ + (cid:1) ◦ e A , ¯ λ − ◦ (cid:0) id K ( H ) ⊗ φ − (cid:1) ◦ e A ) = (cid:0) ¯ λ − ◦ ¯ e K ( H ) ◦ φ + , ¯ λ − ◦ ¯ e K ( H ) ⊗ φ − (cid:1) ∼ ( φ + , φ − ) EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 51 in F ( A ; C ) or, equivalently, (cid:0) ¯ e K ( H ) ◦ φ + , ¯ e K ( H ) ⊗ φ − (cid:1) ∼ (cid:0) ¯ λ ◦ φ + , ¯ λ ◦ φ − (cid:1) in F H ⊗ H ( A ; C ), where F H ⊗ H ( A ; C ) is defined as F ( A ; C ) by replacing H with H ⊗ H . Indeed, by definition of λ , (cid:0) ¯ λ ◦ φ + , ¯ λ ◦ φ − (cid:1) = (cid:0) Ad ( V ) ◦ ¯ e K ( H ) ◦ φ + , Ad ( V ) ◦ ¯ e K ( H ) ◦ φ − (cid:1) .By [JT91, Lemma 1.3.7] there exists a stritly continuous path ( V t ) t ∈ [0 , of isometries in B ( H ⊗ H ) connecting I to V . Thus, (cid:0) Ad ( V t ) ◦ ¯ e K ( H ) ◦ φ + , Ad ( V t ) ◦ ¯ e K ( H ) ◦ φ − (cid:1) is a continuous path in F H ⊗ H ( A ; C ) connecting (cid:0) ¯ e K ( H ) ◦ φ + , ¯ e K ( H ) ⊗ φ − (cid:1) to (cid:0) ¯ λ ◦ φ + , ¯ λ ◦ φ − (cid:1) . This concludes theproof that KK h ( e A ; C ) ◦ G is the identity of KK h ( A ; C ).We now show that G ◦ KK h ( e A ; C ) is the identity of KK h ( K ( H ) ⊗ A ; C ). We have that G ◦ KK h ( A ; C ) is thedefinable group homomorphism induced by the Borel function F ( K ( H ) ⊗ A ; C ) → F ( K ( H ) ⊗ A ; C ) , ( ψ + , ψ − ) (cid:0) ¯ λ − ◦ id K ( H ) ⊗ ( ψ + ◦ e A ) , ¯ λ − ◦ id K ( H ) ⊗ ( ψ − ◦ e A ) (cid:1) .We claim that (cid:0) ¯ λ − ◦ id K ( H ) ⊗ ( ψ + ◦ e A ) , ¯ λ − ◦ id K ( H ) ⊗ ( ψ − ◦ e A ) (cid:1) ∼ ( ψ + , ψ − )in F ( K ( H ) ⊗ A ; C ) or, equivalently (cid:0) id K ( H ) ⊗ ( ψ + ◦ e A ) , id K ( H ) ⊗ ( ψ − ◦ e A ) (cid:1) ∼ (cid:0) ¯ λ ◦ ψ + , ¯ λ − ◦ ψ − (cid:1) = (cid:0) Ad ( V ) ◦ ¯ e K ( H ) ◦ ψ + , Ad ( V ) ◦ ¯ e K ( H ) ◦ ψ − (cid:1) in F H ⊗ H ( K ( H ) ⊗ A ; C ). Indeed, define σ , σ : K ( H ) ⊗ A → K ( H ) ⊗ K ( H ) ⊗ A be the (strict) *-homomorphismsgiven by T ⊗ a T ⊗ e ⊗ a and T ⊗ a e ⊗ T ⊗ a .We can consider their strict extensions ¯ σ , ¯ σ : M ( K ( H ) ⊗ A ) → M ( K ( H ) ⊗ K ( H ) ⊗ A ). Then we have thatid K ( H ) ⊗ ( ψ ± ◦ e A ) = ψ ± ◦ σ : K ( H ) ⊗ A → B ( H ⊗ H )and ¯ e K ( H ) ◦ ψ ± = (cid:0) id K ( H ) ⊗ ψ + (cid:1) ◦ σ : K ( H ) ⊗ A → B ( H ⊗ H ) .We have that ¯ σ = Ad ( U ⊗ ◦ ¯ σ for some unitary U ∈ M ( K ( H ) ⊗ K ( H ) ⊗ A ). Since M ( K ( H ) ⊗ K ( H ) ⊗ A )is connected in the strict topology [JT91, Lemma 1.3.7], we have that (cid:0) id K ( H ) ⊗ ( ψ + ◦ e A ) , id K ( H ) ⊗ ( ψ − ◦ e A ) (cid:1) = (cid:0)(cid:0) id K ( H ) ⊗ ψ + (cid:1) ◦ σ , (cid:0) id K ( H ) ⊗ ψ − (cid:1) ◦ σ (cid:1) ∼ (cid:0)(cid:0) id K ( H ) ⊗ ψ + (cid:1) ◦ σ , (cid:0) id K ( H ) ⊗ ψ − (cid:1) ◦ σ (cid:1) = (cid:0) ¯ e K ( H ) ◦ ψ + , ¯ e K ( H ) ◦ ψ − (cid:1) ∼ (cid:0) Ad ( V ) ◦ ¯ e K ( H ) ◦ ψ + , Ad ( V ) ◦ ¯ e K ( H ) ◦ ψ − (cid:1) .This concludes the proof. (cid:3) Split exactness. Suppose that 0 → A i → B p → C → definably split if p is a split epimorphism in the category of definable groups, namely there exists a definable group homomorphism g : C → B such that p ◦ g is equal to the identity of C . This is equivalent to the assertion that i : A → B isa split monomorphism in the category of definable groups, namely there exists a definable group homomorphism f : B → A such that f ◦ i is equal to the identity of A . In turn, this is equivalent to the assertion that there existsa definable isomorphism γ : B → A ⊕ C that makes the diagram A B CA A ⊕ C C id A γ id C commute.If ( A, J ) is a separable C*-pair such that the exact sequence0 → J → A → A/J → A, J ) is, in particular, semi-split. Thus, there is a corresponding six-term exact sequence in K-homology.This reduces to two definably split exact sequences of definable groups and definable group homomorphisms0 → K p ( J ) → K p ( A ) → K p ( A/J ) → p ∈ { , } . This is the split-exactness property of definable K-homology in the sense of [Cun87].7. A definable Universal Coefficient Theorem In this section we consider a definable version of the Universal Coefficient Theorem for K-homology due to Brown[Bro84], later generalized by Rosenberg and Schochet to KK-theory [RS87]. We also consider the fine structure ofthe definable K-homology groups as in [Sch96] in terms of the notion of filtration for a separable nuclear C*-algebraintroduced therein. As an application, we show that definable K-homology is a complete invariant for UHF C*-algebras up to stable isomorphism, while the same conclusion does not hold for the purely algebraic K-homology.In this section, we assume all the C*-algebras to be separable and nuclear.7.1. Index pairing for K -homology. Suppose that A is a separable, nuclear C*-algebra. Fix p ∈ { , } . Thenone can define a natural definable index pairing K p ( A ) × K p ( A ) → Z , where Z and the countable group K p ( A )are regarded as standard Borel spaces with respect to the trivial Borel structure. Suppose that A is concretelyrealized as a C*-subalgebra of B ( H ) such that the inclusion map A → B ( H ) is an ample representation of A , andlet D ( A ) ⊆ B ( H ) be the corresponding Paschke dual algebra. For p = 1 the pairing is defined by h [ P ] , [ u ] i = Index P H k (cid:0) P ⊕ k uP ⊕ k (cid:1) where k ≥ u ∈ U ( M k ( A + )) is a unitary, P ∈ D ( A ) is a projection, P ⊕ k is the k -fold direct sum of P , P ⊕ k uP ⊕ k ∈ B (cid:0) H k (cid:1) satisfies (cid:0) P ⊕ k uP ⊕ k (cid:1) ∗ (cid:0) P ⊕ k uP ⊕ k (cid:1) ≡ (cid:0) P ⊕ k uP ⊕ k (cid:1) (cid:0) P ⊕ k uP ⊕ k (cid:1) ∗ ≡ P ⊕ k mod K (cid:0) H k (cid:1) ,and Index P ⊕ k H k (cid:0) P ⊕ k uP ⊕ k (cid:1) is its Fredholm index of P ⊕ k uP ⊕ k regarded as a Fredholm operator on P ⊕ k H k ;see [HR00, Definition 7.2.1]. Such a pairing is definable, in the sense that it is induced by a Borel functionZ ( D ( A )) × K ( A ) → Z , considering that the Fredholm index is given by a Borel map; see Section 2.10.The index pairing K ( A ) × K ( A ) → Z is defined by h [ U ] , [ p ] − [ q ] i = Index pH k (cid:0) pU ⊕ k p (cid:1) − Index qH k (cid:0) qU ⊕ k q (cid:1) ,where k ≥ p, q ∈ M k ( A + ) are projections that satisfy p ≡ q mod M k ( A ), pU ⊕ k p ∈ B (cid:0) H k (cid:1) satisfies (cid:0) pU ⊕ k p (cid:1) ∗ (cid:0) pU ⊕ k p (cid:1) ≡ (cid:0) pU ⊕ k p (cid:1) (cid:0) pU ⊕ k p (cid:1) ∗ ≡ p mod K (cid:0) H k (cid:1) ,and Index pH k (cid:0) pU ⊕ k p (cid:1) is the Fredholm index of pU ⊕ k p regarded as a Fredholm operator on pH k and similarly for qU ⊕ k q ; see [HR00, Definition 7.2.3]. Again, this pairing is definable since the Fredholm index is given by a Borelmap.7.2. Extensions of groups. Suppose that C, D are countable abelian groups. A (2-)cocycle on C with coefficientsin D is a function c : C × C → D such that, for every x, y, z ∈ C : • c ( x, y ) + c ( x + y, z ) = c ( x, y + z ) + c ( y, z ); • c ( x, y ) = c ( y, x ).A cocycle is a coboundary if it is of the form ( x, y ) h ( x ) + h ( y ) − h ( x, y ) for some function h : C → D . The setZ ( C, D ) of cocycles on C with coefficients in D is a closed subgroup of the Polish group D C × C endowed with theproduct topology (where D is endowed with the discrete topology). The set B ( C, D ) of coboundaries is a PolishableBorel subgroup of Z ( C, D ). A weak coboundary is a cocycle c such that, for every finite (or, equivalently, for everyfinitely-generated) subgroup S of C , the restriction of c to S × S is a coboundary for S . Weak coboundaries forma closed subgroup B w ( C, D ) of Z ( C, D ), which is in fact the closure of B ( C, D ) inside of Z ( C, D ). EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 53 The group Ext ( C, D ) is the definable group, which is in fact a group with Polish cover (see Remark 1.17), obtainedas the quotient Z ( C, D ) / B ( C, D ); see [BLP20, Section 7]. The pure (or phantom ) Ext group PExt ( C, D ) is thedefinable subgroup of Ext ( C, D ) obtained as B w ( C, D ) / B ( C, D ); see [Sch03, CS98]. We also define Ext w ( C, D )to be the Polish group obtained as the quotient of the Polish group Z ( C, D ) by the closed subgroup B w ( C, D ). Bydefinition, we have a short exact sequence of definable groups0 → PExt ( C, D ) → Ext ( C, D ) → Ext w ( C, D ) → D with coefficients in D gives rise to an extension of C by D , in such a way that two cocycles differ by a coboundary if and only if the corresponding extensions areisomorphic. Furthermore, every extension of C by D arises from a cocycle in this fashion. Explicitly, if0 → D i → E p → C → C by D , the corresponding cocycle c is defined as follows. Fix a right inverse t : C → E for thefunction p : E → C . Then one defines c ( x, y ) := i − ( t ( x ) + t ( y ) − t ( x + y )) ∈ D for x, y ∈ C . Conversely, given acocycle c on C with coefficients in D one can define an extension as above, where E = C × D is endowed with theoperation defined by ( x, y ) + ( x ′ , y ′ ) = ( x + x ′ , c ( x, x ′ ) + y + y ′ ) .The weak coboundaries correspond in this way to extension of C by D that are pure , i.e. such that i ( D ) is a puresubgroup of E ; see [Fuc70, Section V.29].If ( C i ) i ∈ ω is an inductive sequence of finitely-generated abelian groups and C = colim i ∈ ω C i is the correspondinginductive limit (colimit), then the definable Jensen theorem asserts that PExt ( C, D ) is naturally definably isomor-phic to lim i Hom ( C i , D ), and Ext w ( C, D ) is naturally isomorphic as a Polish group to lim i Ext ( C i , D ); see [BLP20,Theorem 7.4] and [Sch03, Theorem 6.1].7.3. The Universal Coefficient Theorem. Suppose that A is a separable, nuclear C*-algebra. The definableindex pairing K ( A ) × K ( A ) → Z induces a definable homomorphismIndex A : K ( A ) → Hom (K ( A ) , Z ) ,where we adopt the notation from [HR00, Definition 7.2.3]. Recall that K ( A ) is defined as Ext( A + ) − where A + is the unitization of A . The definable homomorphism Index A : Ext ( A + ) − → Hom (K ( A ) , Z ) can be equivalentlydescribed as follows; see [RS81]. Let τ be an injective unital extension0 → K ( H ) → E → A + → A + by K ( H ). Then τ gives rise to a six-term exact sequence in K-theoryK ( K ( H )) = Z K ( E ) K ( A + )K ( A + ) = K ( A ) K ( E ) K ( K ( H )) = { } ∂ ∂ The group homomorphism K ( A ) → Z induced by τ in the diagram above depends only on the corresponding ele-ment [ τ ] of Ext ( A + ) − , and it is equal to Index A ([ τ ]). As in [HR00, Definition 7.6.7], we let ◦ K ( A ) be the definablesubgroup of K ( A ) obtained as the kernel of the index homomorphism Index A : K ( A ) → Hom (K ( A ) , Z ).There is also a definable group homomorphism κ A : ◦ K ( A ) → Ext (K ( A ) , Z ), defined as follows. Suppose that τ is an injective unital extension of A + by K ( H ) as above, such that moreover [ τ ] ∈ ◦ K ( A ). Then the six-termexact sequence above reduces to a short exact sequence0 → K ( K ( H )) = Z → K ( E ) → K (cid:0) A + (cid:1) → ( A + ) , Z ), which in turn defines an element of Ext (K ( A ) , Z ) via the inclusionK ( A ) → K ( A + ). This element κ A ([ τ ]) of Ext (K ( A ) , Z ) depends only on the class [ τ ] in Ext ( A + ) − of theextension τ . This gives a group homomorphism κ A : ◦ K ( A ) → Ext (K ( A ) , Z ), [ τ ] κ A ([ τ ]), which is easilyseen to be definable. In a similar fashion, by replacing A with its suspension, one can define a definable grouphomomorphism Index A : K ( A ) → Hom (K ( A ) , Z ) with kernel ◦ K ( A ), and a definable group homomorphism κ A : ◦ K ( A ) → Ext (K ( A ) , Z ) .We recall the following definition of a C*-algebra satisfying the Universal Coefficient Theorem (UCT); see [RS81,Definition 4.4]. Definition 7.1. A separable C*-algebra A is said to satisfy the Universal Coefficient Theorem (UCT) for C , orthe pair ( A, C ) satisfies the UCT, if for p ∈ { , } the group homomorphisms Index A : K p ( A ) → Hom (K p ( A ) , Z )is surjective, and the group homomorphism κ A : Ker ( γ A ) = ◦ K p ( A ) → Ext (K − p ( A ) , Z ) is an isomorphism.It is proved in [Bro84] that all the separable nuclear C*-algebras in the so-called bootstrap class satisfy theUCT for C ; see also [Bro75]. In fact one can more generally consider the UCT for B , where B is any separableC*-algebra, defined in terms of Kasparov’s KK-groups; see [RS87]. It is unknown whether there exists a separablenuclear C*-algebra that does not satisfy the UCT.7.4. Weak and asymptotic K -homology groups. We now recall the notion of a filtration (or KK-filtration) fora separable nuclear C*-algebra as in [Sch96, Definition 1.4], and we define the weak and asymptotic K-homologygroups for C*-algebras with a filtration. Definition 7.2. Suppose that A is a separable, nuclear C*-algebra. An inductive sequence ( A n , η n ) n ∈ ω of separable,nuclear C*-algebras is a filtration of A if: • for every n ∈ ω , A n satisfies the Universal Coefficient Theorem for C (as in Definition 7.1); • for every n ∈ ω and p ∈ { , } , K p ( A n ) is a finitely generated group; • A is KK-equivalent to the inductive limit of the sequence ( A n , η n ) n ∈ ω . Remark 7.3. A slightly more restrictive definition is considered in [Sch96, Definition 1.4], where the C*-algebras A n are supposed to commutative.We let C be the category that has separable, nuclear C*-algebras with a filtration as objects, and *-homomorphismsas morphisms.Suppose that A is a separable, nuclear C*-algebra with a filtration ( A n ) n ∈ ω . Then the inductive limit colim n A n of the sequence ( A n ) n ∈ ω satisfies the UCT for C by [Sch96, Theorem 4.1], whence A satisfies the UCT as well.Thus, the definable group homomorphism κ A : ◦ K p ( A ) = Ker (Index A ) → Ext (K p ( A ) , Z ) is an isomorphism.After replacing A with colim n A n we can assume that A = colim n A n . Notice that, as K ( A n ) and K ( A n ) arefinitely-generated and A n satisfies the UCT for C , it follows that K ( A n ) and K ( A n ) are countable groups.We define the weak K-homology group K p w ( A ) to be Polish group lim n K p ( A n ). The assignment A K p w ( A )defines a homotopy-invariant functor from C to the category of Polish groups. The weak K-homology group K ( A )is isomorphic to the group KL ( A, C ) from [Rør95, Section 4]; see also [RS02, 2.4.8] and [Sch96, Corollary 3.8].A description of K ( A ) in terms of the sum K ( A ) of all the K-theory groups of A in all degrees and all cycliccoefficient groups is obtained in [DL96]; see also [Sch96, Theorem 3.10].We have a canonical surjective definable homomorphism K p ( A ) → K p w ( A ) as in Milnor’s exact sequence. Wedefine the asymptotic K-homology group K p ∞ ( A ) to be the kernel of such a definable homomorphism. As A satisfiesthe UCT for C , the definable isomorphism κ A : ◦ K p ( A ) → Ext (K − p ( A ) , Z ) is an isomorphism. Since, for every n ∈ ω , A n satisfies the UCT for C , K p ∞ ( A ) ⊆ ◦ K p ( A ) is equal to the inverse image of PExt (K − p ( A ) , Z ) under κ A . In particular, this shows that K p ∞ ( A ) does not depend on the choice of the filtration for A . The assignment A K p ∞ ( A ) defines a homotopy-invariant functor from C to the category of definable groups. As noticed above,K p ∞ ( A ) is naturally definably isomorphic to PExt (K − p ( A ) , Z ). We also have that K − p ( A ) = colim n K − p ( A n ),and hence K p ∞ ( A ) is definably isomorphic to lim n Hom (K − p ( A n ) , Z ) by the definable Jensen theorem [BLP20,Theorem 7.4]. Lemma 7.4. Suppose that A is a separable, nuclear C*-algebra, and ( A n , η n ) n ∈ ω is a filtration of A . Thenthe definable homomorphism K p ( A ) → K p w ( A n ) has a definable right inverse K p w ( A n ) → K p ( A ) , which is notnecessarily a group homomorphism.Proof. After replacing A with it suspension, we can assume that p = 1. Furthermore, after replacing A withcolim n A n , we can assume that A = colim n A n . Finally, after replacing A with A + and A n with A + n , we can assume EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 55 that A and A n for n ∈ ω are unital, and η n : A n → A n +1 is a unital *-homomorphism. In this case, we have thatK ( A ) = Ext ( A ) − and K ( A n ) = Ext ( A n ) − for n ∈ ω . We need to show that the definable group homomorphismExt ( A ) − → lim n Ext ( A n ) − has a definable right inverse, which is not necessarily a group homomorphism. Recallthat Ext( A ) − is the quotient of the Polish space E ( A ) of representatives of injective, unital extensions of A by theequivalence relation ≈ as in Section 4.1.Fix, for every ℓ ∈ ω an enumeration ( x ( ℓ ) n ) n ∈ ω of Ext ( A ℓ ) − . For ℓ < ℓ define the bonding map η ( ℓ ,ℓ ) = η ℓ − ◦ · · · ◦ η ℓ : A ℓ → A ℓ ,and set η ( ℓ,ℓ ) = id A ℓ for ℓ ∈ ω . Define η ( ∞ ,ℓ ) : A ℓ → A to be the canonical map. Let also p ( ℓ ,ℓ ) : Ext ( A ℓ ) − → Ext ( A ℓ ) − be the group homomorphism induced by thebonding map η ( ℓ ,ℓ ) : A ℓ → A ℓ . Then an element of lim n Ext ( A n ) − is a sequence ( x ( ℓ ) n ℓ ) ℓ ∈ ω such that, for ℓ < ℓ , p ( ℓ ,ℓ ) ( x ℓ n ℓ ) = x ℓ n ℓ . For every ℓ, n ∈ ω fix ϕ ( ℓ ) n ∈ E ( A ℓ ) such that [ ϕ ( ℓ ) n ] = x ( ℓ ) n . For every ℓ ∈ ω and n, m ∈ ω suchthat p ( ℓ − ,ℓ ) ( x ( ℓ ) n ) = p ( ℓ − ,ℓ ) ( x ( ℓ ) m ) fix U ( ℓ ) n,m ∈ U ( H ) such that Ad( U ( ℓ ) n,m ) ◦ ϕ ( ℓ ) n ◦ η ℓ − = ϕ ( ℓ ) m ◦ η ℓ − . If ( x ( ℓ ) n ℓ ) n ∈ ω isan element of lim n Ext ( A n ) − , then setting ψ ( ℓ ) := Ad( U ( ℓ ) n ℓ ,n ℓ − U ( ℓ − n ℓ − ,n ℓ − · · · U (1) n n ) ◦ ϕ ( ℓ ) n ℓ ∈ E ( A ℓ ), one obtains asequence (cid:0) ψ ( ℓ ) (cid:1) ℓ ∈ ω such that ψ ( ℓ ) ◦ η ℓ − = ψ ( ℓ − for every ℓ > 0. Therefore, setting ψ = colim ℓ ψ ( ℓ ) : A → B ( H )defines an element of E ( A ) such that (cid:2) ψ ◦ η ( ∞ ,ℓ ) (cid:3) = x ℓn ℓ for every ℓ ∈ ω , and hence the image of [ ψ ] ∈ Ext ( A ) − under the definable homomorphism Ext ( A ) − → lim n Ext ( A n ) − is equal to (cid:0) x ℓn ℓ (cid:1) ℓ ∈ ω . This construction describesa definable function lim n Ext ( A n ) − → Ext ( A ) − , which is a right inverse for Ext ( A ) − → lim n Ext ( A n ) − . Thisconcludes the proof. (cid:3) Suppose that A is a separable, nuclear C*-algebra with a filtration ( A n , η n ) n ∈ ω . The index homomorphismsIndex A n : K p ( A n ) → Hom (K p ( A n ) , Z )for n ∈ ω induce a continuous group homomorphismK p w ( A ) → Hom (K p ( A ) , Z ) = lim n Hom (K p ( A n ) , Z ) .Similarly the definable group homomorphisms κ − A n : Ext (K p ( A n ) , Z ) → K p ( A n )for n ∈ ω induce a definable group homomorphismlim n Ext (K p ( A n ) , Z ) = Ext w (K p ( A ) , Z ) → K p w ( A ) . This gives a short exact sequence of definable groups0 → Ext w (K p ( A ) , Z ) → K p w ( A ) → Hom (K p ( A ) , Z ) → w , we also have a short exact sequence of definable groups0 → PExt (K p ( A ) , Z ) → Ext (K p ( A ) , Z ) → Ext w (K p ( A ) , Z ) → p ( A ) , Z ) → Ext (K p ( A ) , Z ) is the inclusion map and Ext (K p ( A ) , Z ) → Ext w (K p ( A ) , Z ) is thequotient map. Proposition 7.5. Suppose that A is a separable, nuclear C*-algebra with a filtration and p ∈ { , } . If K p ( A ) istorsion-free, then K p ∞ ( A ) is naturally isomorphic to Ext (K − p ( A ) , Z ) , and K p w ( A ) is naturally isomorphic as aPolish group to Hom (K p ( A ) , Z ) .Proof. Since K p ( A ) is torsion-free, we have that PExt (K p ( A ) , Z ) = Ext (K p ( A ) , Z ). Therefore,K p ∞ ( A ) ∼ = PExt (K p ( A ) , Z ) = Ext (K p ( A ) , Z ) .From the exact sequence0 → PExt (K p ( A ) , Z ) → Ext (K p ( A ) , Z ) → Ext w (K p ( A ) , Z ) → w (K p ( A ) , Z ) = { } . From this and the exact sequence0 → Ext w (K p ( A ) , Z ) → K p w ( A ) → Hom (K p ( A ) , Z ) → p w ( A ) ∼ = Hom (K p ( A ) , Z ) .This concludes the proof. (cid:3) Corollary 7.6. Suppose that A is a separable, nuclear C*-algebra with a filtration and p ∈ { , } is such that K p ( A ) is a finite-rank torsion-free abelian group and K − p ( A ) is trivial. We can write K p ( A ) = Λ ⊕ Λ ′ where Λ ′ is finitely-generated and Λ has no nonzero finitely-generated direct summand. Then K p ( A ) ∼ = Hom (K p ( A ) , Z ) ∼ = Hom (Λ ′ , Z ) and K − p ( A ) ∼ = Ext (K p ( A ) , Z ) ∼ = Ext (Λ , Z ) as definable groups.Proof. After replacing A with SA , we can assume that p = 0. We have thatK ∞ ( A ) ∼ = PExt (K ( A ) , Z ) ∼ = { } .Therefore, K ( A ) ∼ = K ( A ) ∼ = Hom (K ( A ) , Z ) ∼ = Hom (Λ ′ , Z ) .Similarly, we have that K ( A ) ∼ = Hom (K ( A ) , Z ) ∼ = { } and hence, since K ( A ) is torsion-free,K ( A ) ∼ = K ∞ ( A ) ∼ = PExt (K ( A ) , Z ) ∼ = Ext (K ( A ) , Z ) ∼ = Ext (Λ , Z ) .This concludes the proof. (cid:3) Corollary 7.7. Suppose that p ∈ { , } and A, B are separable, nuclear C*-algebras with a filtration, such that K p ( A ) and K p ( B ) are finite-rank torsion-free abelian groups, and K − p ( A ) and K − p ( B ) are trivial. Then thefollowing assertions are equivalent:(1) K i ( A ) and K i ( B ) are definably isomorphic for i ∈ { , } ;(2) K p ( A ) and K p ( B ) are isomorphic.If furthermore K p ( A ) and K p ( B ) have no nonzero finitely-generated direct summand, then the following asser-tions are equivalent:(1) K − p ( A ) and K − p ( B ) are definably isomorphic;(2) K p ( A ) and K p ( B ) are isomorphic.Proof. After passing to the suspension, we can assume that p = 0. Since K ( A ) and K ( B ) are finite-rank torsion-free abelian groups, we can write K ( A ) = Λ A ⊕ Λ ′ A K ( B ) = Λ B ⊕ Λ ′ B where Λ A , Λ B have no nonzero finitely-generated direct summand, and Λ ′ A , Λ ′ B are finitely-generated. Then we havethat K ( A ) ∼ = K ( B ) if and only if Λ A ∼ = Λ B and Λ ′ A ∼ = Λ ′ B . We have that Λ ′ A ∼ = Λ ′ B if and only ifHom (Λ ′ A , Z ) ∼ = Hom (Λ ′ B , Z )Furthermore, by [BLP20, Corollary 7.6], we have that Λ A ∼ = Λ B if and only if Ext (Λ A , Z ) and Ext (Λ B , Z ) aredefinably isomorphic. The conclusion thus follows from Corollary 7.6. (cid:3) We now show that Corollary 7.7 does not hold if K p ( A ) and K p ( B ) are merely asked to be isomorphic, ratherthan definably isomorphic; see Theorem 7.8. EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 57 Stable isomorphism of UHF algebras. Recall that a uniformly hyperfinite (UHF) C*-algebra is an infinite-dimensional separable unital C*-algebra that is the limit of an inductive sequence of full matrix algebras [Dav96,Example III.5.1]. Since finite-dimensional C*-algebras are nuclear, satisfy the UCT for C , and have finitely-generatedK and K groups, UHF C*-algebras are nuclear and have a filtration. If A is a UHF C*-algebra, then K ( A ) is arank 1 torsion-free abelian group that is not isomorphic to Z , while K ( A ) is trivial. Given a rank 1 torsion-freeabelian group Λ that is not isomorphic to Z , there exists a UHF C*-algebra A Λ such that K ( A Λ ) ∼ = Λ. ByProposition 7.5, we have that K ( A Λ ) is definably isomorphic to Ext (Λ , Z ), while K ( A Λ ) is trivial.Recall that a rank 1 torsion-free abelian group is an abelian group that is isomorphic to a subgroup of Q . Givena torsion-free group Λ and a prime number p , one defines its p -corank rank p Λ to be the dimension of Λ /p Λ asa Z /p Z -vector space. As a particular instance of [BLP20, Theorem A.7] we have that, given rank 1 torsion-freeabelian groups Λ , Λ ′ , Ext (Λ , Z ) and Ext (Λ ′ , Z ) are isomorphic as discrete groups if and only if rank p Λ = rank p Λ ′ for every prime p . It easily follows that there exists an uncountable family (Λ i ) i ∈ R of pairwise nonisomorphic rank1 torsion-free abelian groups such that Ext (Λ i , Z ) and Ext (Λ j , Z ) are isomorphic as discrete groups for i, j ∈ R .The following result is an immediate consequence of these observations together with Corollary 7.7 and theElliott classification of approximately finite-dimensional (AF) C*-algebras [Ell76], or Glimm’s classification of UHFC*-algebras [Gli60]; see also [RLL00, Chapter 7]. Recall that two separable C*-algebras A, B are stably isomorphic (or, equivalently, Morita-equivalent; see [RW98, Definition Theorem 5.55]) if A ⊗ K ( H ) ∼ = B ⊗ K ( H ), where K ( H )is the C*-algebra of compact operators on the separable infinite-dimensional Hilbert space. Theorem 7.8. Definable K is a complete invariant for UHF C*-algebras up to stable isomorphism. In contrast,there exists an un uncountable family of pairwise non stably isomorphic UHF C*-algebras whose K -groups areisomorphic as discrete groups (but not definably isomorphic).Proof. It follows from the classification of AF C*-algebras by K-theory that the (unordered) K -group is a completeinvariant for UHF C*-algebras up to stable isomorphism; see [Dav96, Chapter IV]. From this and Corollary 7.7, itfollows that the definable K -group is also a complete invariant for UHF C*-algebras up to stable isomorphism.If, adopting the notations above, (Λ i ) i ∈ R is an uncountable family of pairwise nonisomorphic rank 1 torsion-free abelian groups not isomorphic to Z such that Ext (Λ i , Z ) and Ext (Λ j , Z ) are isomorphic as discrete groupsfor i, j ∈ R , then ( A Λ i ) i ∈ R is an uncountable family of pairwise non stably isomorphic UHF C*-algebras whoseK -groups are isomorphic as discrete groups but not definably isomorphic. (cid:3) Definable K -homology of compact metrizable spaces In this section, we consider definable K-homology of compact metrizable spaces, which can be seen as a partic-ular instance of definable K-homology when restricted to unital, commutative, separable C*-algebras. As anotherapplication of the definable Universal Coefficient Theorem, we show that definable K-homology of compact metriz-able spaces is a finer invariant than its purely algebraic version, even when restricted to connected 1-dimensionalsubspaces of R .8.1. K -homology and topological K -theory of spaces. The notion (definable) of K-homology for compactmetrizable spaces is obtained as a particular instance of the corresponding notion for separable C*-algebras, byconsidering the contravariant functor X C ( X ) assigning to a compact metrizable space the separable unitalC*-algebra C ( X ) of continuous complex-valued functions on X . Thus, if X is a compact metrizable space, its definable K-homology groups are given by K p ( X ) := K p ( C ( X ))for p ∈ { , } ; see [HR00, Chapter 7]. The reduced definable K-homology groups are similarly defined by˜K p ( X ) := ˜K p ( C ( X )) .In particular, one sets Ext ( X ) := Ext ( C ( X ))and ˜K ( X ) = ˜K ( C ( X )) . Notice that, by definition, ˜K ( X ) = ˜K ( C ( X )) = Ext ( C ( X )) = Ext ( X ) .Similarly, the topological K -theory groups of X can be defined in terms of the K-theory of C ( X ) by settingK p ( X ) := K p ( C ( X )) ;see [RLL00, 3.3.7]. Equivalently, the topological K-groups can be defined in terms of vector bundles over X ; see[Kar08, Chapter II] and [WO93, Chapter 13]. One can also define the reduced K-group ˜K p ( X ) to be the quotient ofK p ( X ) by the subgroup obtained as the image of K p ( {∗} ) under the homomorphism induced by the map X → {∗} .(Notice that K ( {∗} ) is trivial and K ( {∗} ) ∼ = Z .)8.2. The Universal Coefficient Theorem. Recall that a compact polyhedron is a compact metrizable space thatis obtained as the topological realization of a finite simplicial complex; see [MS82, Appendix 1]. (In the following, weassume that all the polyhedra are compact.) The topological K-groups of a polyhedron are finitely-generated [HR00,Proposition 7.14]. Furthermore, if P is a polyhedron, then it can be proved by induction on the number of simplicesof the corresponding simplicial complex that the unital C*-algebra C ( P ) satisfies the UCT for C [Bro75, Bro84].If X is a compact metrizable space, then one can write X as the (inverse) limit of a tower ( X n ) n ∈ ω of compactpolyhedra [MS82, Section I.6]. Such a tower, called a polyhedral resolution of X in [MS82], can be obtained byconsidering the topological realizations of the nerves of a sequence of finite open covers of X that is cofinal in theordered set of finite open covers of X . If ( X n ) is a polyhedral resolution for X , then ( C ( X n )) n ∈ ω is a filtration for C ( X ) in the sense of Definition 7.2. Thus, one can consider the weak K-homology groupK w p ( X ) := K p w ( C ( X )) = lim n K p ( X n )and the asymptotic K-homology groupsK ∞ p ( X ) := K p ∞ ( C ( X )) ∼ = PExt (cid:0) K − p ( X ) , Z (cid:1) .We can also consider their reduced versions, by letting ˜K w p ( X ) be the kernel of the definable group homomorphismK w p ( X ) → K w p ( {∗} ) induced by the map X → {∗} , and similarly for ˜K ∞ p ( X ). It is then easy to see that˜K w p ( X ) = lim n ˜K p ( X n )and ˜K ∞ p ( X ) ∼ = K ∞ p ( X ) .By definition, we have definable short exact sequences0 → K ∞ p ( X ) → K p ( X ) → K w p ( X ) → → ˜K ∞ p ( X ) → ˜K p ( X ) → ˜K w p ( X ) → Proposition 8.1. Suppose that X is a compact metrizable space and p ∈ { , } . If ˜K p ( X ) is torsion-free, then ˜K ∞ p ( X ) is naturally definably isomorphic to Ext( ˜K − p ( X ) , Z ) , and ˜K w p ( X ) is naturally isomorphic to Hom( ˜K p ( X ) , Z ) . Corollary 8.2. Suppose that X is a compact metrizable space and p ∈ { , } is such that ˜K p ( X ) is a finite-ranktorsion-free abelian group and ˜K − p ( X ) is trivial. We can write ˜K p ( X ) = Λ ⊕ Λ ′ where Λ ′ is finitely-generated and Λ has no nonzero finitely-generated direct summand. Then ˜K w p ( X ) ∼ = Hom( ˜K p ( X ) , Z ) ∼ = Hom (Λ ′ , Z ) and ˜K ∞ − p ( A ) ∼ = Ext( ˜K p ( X ) , Z ) ∼ = Ext (Λ , Z ) as definable groups. EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 59 Corollary 8.3. Suppose that p ∈ { , } , and X, Y are compact metrizable spaces, such that ˜K p ( X ) and ˜K p ( Y ) arefinite-rank torsion-free abelian groups, and ˜K − p ( X ) and ˜K − p ( Y ) are trivial. Then the following assertions areequivalent:(1) ˜K i ( X ) and ˜K i ( Y ) are definably isomorphic for i ∈ { , } ;(2) ˜K p ( A ) and ˜K p ( B ) are isomorphic.If furthermore K p ( X ) and K p ( Y ) have no nonzero finitely-generated direct summand, then the following asser-tions are equivalent:(1) ˜K − p ( X ) and ˜K − p ( Y ) are definably isomorphic;(2) ˜K p ( A ) and ˜K p ( B ) are isomorphic. Solenoids. A (1-dimensional) solenoid is a compact metrizable space X that is homeomorphic to a 1-dimensionalcompact connected abelian group other than T . Thus, if Λ is a rank 1 torsion-free abelian group (or, equivalently,a subgroup of Q ) other than Z , then its Pontryagin dual group X Λ := Λ ∗ is a solenoid, and every solenoid arises inthis fashion (up to homeomorphism). When Λ = Z [1 /p ] for some prime number p , then the corresponding solenoid X Λ is called the p -adic solenoid. A solenoid X can be realized as a compact subset of R (but not of R ) [ES52,Exercise VIII.E]; see also [JWZZ11, JWZ08, Bog88b, Bog88a]. Solenoids were originally considered by Vietoris[Vie27] and van Danztig [vD32]. They arise in the context of dynamical systems, and they provided in the work ofSmale the first examples of attractors of dynamical systems that are strange [Rue06, Sma67, Wil74].If T is the circle, then one has that ˜K ( T ) = Z and ˜K ( T ) = { } . Furthermore, if ϕ : T → T is a continuousmap of degree n ∈ Z , then the induced map ϕ ∗ : ˜K ( T ) → ˜K ( T ) is given by x nx . It follows easily fromthis that, if Λ is a subgroup of Q , then ˜K ( X Λ ) ∼ = Λ and ˜K ( X Λ ) ∼ = { } . Thus, by Proposition 8.1, we have that˜K ( X Λ ) ∼ = Ext (Λ , Z ) and ˜K ( X Λ ) ∼ = { } as definable groups. (The reduced K-homology of p -adic solenoids is alsocomputed in [KS77, Theorem 6.8].) As in the proof of Theorem 7.8, we have the following. Theorem 8.4. Definable ˜K is a complete invariant for -dimensional solenoids up to homeomorphism. In contrast,there exist uncountably many pairwise non homeomorphic -dimensional solenoids whose ˜K -groups are isomorphicas discrete groups (but not definably isomorphic).Proof. If Λ is a 1-dimensional solenoid, then ˜K ( X Λ ) ∼ = Λ and ˜K ( X Λ ) ∼ = { } . It follows from this and Corollary8.3 that definable ˜K is a complete invariant for 1-dimensional solenoids up to homeomorphism.If (Λ i ) i ∈ R is an uncountable family of pairwise nonisomorphic rank 1 torsion-free abelian groups such thatExt (Λ i , Z ) and Ext (Λ j , Z ) are isomorphic as discrete groups for i, j ∈ R as in Section 7.5, then ( X Λ i ) i ∈ R is anuncountable family of pairwise non homeomorphic solenoids whose ˜K -groups are isomorphic as discrete groups butnot definably isomorphic. (cid:3) References [Ahl78] Lars V. Ahlfors, Complex analysis , third ed., McGraw-Hill Book Co., New York, 1978, An introductionto the theory of analytic functions of one complex variable, International Series in Pure and AppliedMathematics.[AM15] Hiroshi Ando and Yasumichi Matsuzawa, The Weyl–von Neumann theorem and Borel complexity ofunitary equivalence modulo compacts of self-adjoint operators , Proceedings of the Royal Society of Ed-inburgh, Section A (2015), no. 06, 1115–1144.[And20] Esteban Andruchow, A note on geodesics of projections in the Calkin algebra , arXiv:2004.01158 (2020).[APT73] Charles A. Akemann, Gert K. Pedersen, and Jun Tomiyama, Multipliers of C*-algebras , Journal ofFunctional Analysis (1973), 277–301.[Are46] Richard Arens, Topologies for homeomorphism groups , American Journal of Mathematics (1946),593–610.[Arv77] William Arveson, Notes on extensions of C*-algebras , Duke Mathematical Journal (1977), no. 2,329–355.[Ati70] Michael F. Atiyah, Global theory of elliptic operators , Proc. Internat. Conf. on Functional Analysis andRelated Topics (Tokyo, 1969), Univ. of Tokyo Press, Tokyo, 1970, pp. 21–30. [Ati89] M. F. Atiyah, K -theory , second ed., Advanced Book Classics, Addison-Wesley Publishing Company,Advanced Book Program, Redwood City, CA, 1989.[BDF73] Lawrence G. Brown, Ronald G. Douglas, and Peter A. Fillmore, Extensions of C*-algebras, operatorswith compact self-commutators, and K -homology , Bulletin of the American Mathematical Society (1973), 973–978.[BDF77] , Extensions of C*-algebras and K -homology , Annals of Mathematics. Second Series (1977),no. 2, 265–324.[BK96] Howard Becker and Alexander S. Kechris, The descriptive set theory of Polish group actions , LondonMathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996.[Bla86] Bruce Blackadar, K-theory for operator algebras , Mathematical Sciences Research Institute Publications,vol. 5, Springer-Verlag, New York, 1986.[Bla98] , K-theory for operator algebras , second ed., Mathematical Sciences Research Institute Publica-tions, vol. 5, Cambridge University Press, Cambridge, 1998.[Bla06] , Operator Algebras , Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin,2006.[BLP] Jeffrey Bergfalk, Martino Lupini, and Aristotelis Panagiotopoulos, The definable content of homologicalinvariants II: cohomology and homotopy classification , in preparation.[BLP20] , The definable content of homological invariants I: Ext & lim , arXiv:2008.08782 (2020).[BO08] Nathanial P. Brown and Narutaka Ozawa, C*-algebras and finite-dimensional approximations , GraduateStudies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008.[Bog88a] M´aty´as Bogn´ar, On embedding of locally compact abelian topological groups in Euclidean spaces. I , ActaMathematica Hungarica (1988), no. 3-4, 371–399.[Bog88b] , On embedding of locally compact abelian topological groups in Euclidean spaces. II , Acta Math-ematica Hungarica (1988), no. 1-2, 101–131.[Bro75] Lawrence G. Brown, Operator algebras and algebraic K -theory , Bulletin of the American MathematicalSociety (1975), no. 6, 1119–1121, Publisher: American Mathematical Society.[Bro84] , The universal coefficient theorem for Ext and quasidiagonality , Operator algebras and grouprepresentations, Vol. I (Neptun, 1980), Monogr. Stud. Math., vol. 17, Pitman, Boston, MA, 1984,pp. 60–64.[BY08] Ita¨ı Ben Yaacov, Topometric spaces and perturbations of metric structures , Logic and Analysis (2008),no. 3, 235–272.[BYBM13] Ita¨ı Ben Yaacov, Alexander Berenstein, and Julien Melleray, Polish topometric groups , Transactions ofthe American Mathematical Society (2013), no. 7, 3877–3897.[BYM15] Ita¨ı Ben Yaacov and Julien Melleray, Grey subsets of polish spaces , The Journal of Symbolic Logic (2015), no. 4, 1379–1397.[BYU10] Ita¨ı Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability , Transactionsof the American Mathematical Society (2010), no. 10, 5213–5259.[CE76] Man-Duen Choi and Edward G. Effros, The completely positive lifting problem for C*-algebras , Annalsof Mathematics (1976), no. 3, 585–609.[CS98] J. Daniel Christensen and Neil P. Strickland, Phantom maps and homology theories , Topology. AnInternational Journal of Mathematics (1998), no. 2, 339–364.[Cun83] Joachim Cuntz, Generalized homomorphisms between C*-algebras and KK -theory , Dynamics and pro-cesses (Bielefeld, 1981), Lecture Notes in Math., vol. 1031, Springer, Berlin, 1983, pp. 31–45.[Cun84] , K -theory and C*-algebras , Algebraic K -theory, number theory, geometry and analysis (Biele-feld, 1982), Lecture Notes in Math., vol. 1046, Springer, Berlin, 1984, pp. 55–79.[Cun87] , A new look at KK -theory , K -Theory (1987), no. 1, 31–51.[Dad00] Marius Dadarlat, Approximate unitary equivalence and the topology of Ext( A, B ), C*-algebras (M¨unster,1999), Springer, Berlin, 2000, pp. 42–60.[Dad05] , On the topology of the Kasparov groups and its applications , Journal of Functional Analysis (2005), no. 2, 394–418.[Dav96] Kenneth R. Davidson, C*-algebras by Example , Fields Institute Monographs, vol. 6, American Mathe-matical Society, Providence, RI, 1996. EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 61 [DE02] Marius Dadarlat and Søren Eilers, On the classification of nuclear C*-algebras , Proceedings of theLondon Mathematical Society. Third Series (2002), no. 1, 168–210.[DL96] Marius Dadarlat and Terry A. Loring, A universal multicoefficient theorem for the Kasparov groups ,Duke Mathematical Journal (1996), no. 2, 355–377.[Eff65] Edward G. Effros, Transformation Groups and C*-algebras , Annals of Mathematics (1965), no. 1,38–55.[EFP + 13] George A. Elliott, Ilijas Farah, Vern I. Paulsen, Christian Rosendal, Andrew S. Toms, and AsgerT¨ornquist, The isomorphism relation for separable C*-algebras , Mathematical Research Letters (2013), no. 6, 1071–1080.[EK01] George A. Elliott and Dan Kucerovsky, An abstract Voiculescu-Brown-Douglas-Fillmore absorptiontheorem , Pacific Journal of Mathematics (2001), no. 2, 385–409.[Ell76] George A Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensionalalgebras , Journal of Algebra (1976), no. 1, 29–44.[Ell77] George A. Elliott, The Mackey Borel structure on the spectrum of an approximately finite-dimensionalseparable C*-algebra , Transactions of the American Mathematical Society (1977), 59–68.[ES52] Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology , Princeton University Press,Princeton, New Jersey, 1952.[Far11] Ilijas Farah, All automorphisms of the Calkin algebra are inner , Annals of Mathematics (2011),no. 2, 619–661.[Far12] , A dichotomy for the Mackey Borel structure , Proceedings of the 11th Asian Logic Conference,World Scientific Publishing Co. Inc., 2012, pp. 86–93.[Far19] , Combinatorial set theory of C*-algebras , Springer Monographs in Mathematics, Springer,Cham, 2019.[FTT12] Ilijas Farah, Andrew Toms, and Asger T¨ornquist, The descriptive set theory of C*-algebra invariants ,International Mathematics Research Notices (2012), 5196–5226.[FTT14] Ilijas Farah, Andrew S. Toms, and Asger T¨ornquist, Turbulence, orbit equivalence, and the classificationof nuclear C*-algebras , Journal f¨ur die reine und angewandte Mathematik (2014), 101–146.[Fuc70] L´aszl´o Fuchs, Infinite abelian groups. Vol. I , Pure and Applied Mathematics, Vol. 36, Academic Press,New York-London, 1970.[Gao09] Su Gao, Invariant Descriptive Set Theory , Pure and Applied Mathematics, vol. 293, CRC Press, BocaRaton, FL, 2009.[Gha15] Saeed Ghasemi, Isomorphisms of quotients of FDD-algebras , Israel Journal of Mathematics (2015),no. 2, 825–854.[GL16] Eusebio Gardella and Martino Lupini, Conjugacy and cocycle conjugacy of automorphisms of O arenot Borel , M¨unster Journal of Mathematics (2016), no. 1, 93–118.[Gli60] James G. Glimm, On a certain class of operator algebras , Transactions of the American MathematicalSociety (1960), no. 2, 318–340.[Gli61] James Glimm, Type I C*-algebras , Annals of Mathematics. Second Series (1961), 572–612.[Hig87] Nigel Higson, A characterization of KK -theory , Pacific Journal of Mathematics (1987), no. 2,253–276.[Hig95] , C*-algebra extension theory and duality , Journal of Functional Analysis (1995), no. 2,349–363.[HR00] Nigel Higson and John Roe, Analytic K -homology , Oxford Mathematical Monographs, Oxford Univer-sity Press, Oxford, 2000.[I´or80] Val´eria B. de Magalhaes I´orio, Hopf C*-algebras and locally compact groups , Pacific Journal of Mathe-matics (1980), no. 1, 75–96.[JT91] Kjeld Knudsen Jensen and Klaus Thomsen, Elements of KK -theory , Mathematics: Theory & Applica-tions, Birkh¨auser Boston, Inc., Boston, MA, 1991.[JWZ08] Boju Jiang, Shicheng Wang, and Hao Zheng, No embeddings of solenoids into surfaces , Proceedings ofthe American Mathematical Society (2008), no. 10, 3697–3700.[JWZZ11] Boju Jiang, Shicheng Wang, Hao Zheng, and Qing Zhou, On tame embeddings of solenoids into -space ,Fundamenta Mathematicae (2011), no. 1, 57–75. [Kar08] Max Karoubi, K -theory , Classics in Mathematics, Springer-Verlag, Berlin, 2008.[Kas75] Gennadi G. Kasparov, Topological invariants of elliptic operators. I. K -homology , Izv. Akad. Nauk SSSRSer. Mat. (1975), no. 4, 796–838.[Kec94] Alexander S. Kechris, Countable sections for locally compact group actions. II , Proceedings of the Amer-ican Mathematical Society (1994), no. 1, 241–247.[Kec95] , Classical descriptive set theory , Graduate Texts in Mathematics, vol. 156, Springer-Verlag, NewYork, 1995.[Kec98] Alexander Kechris, The descriptive classification of some classes of C*-algebras , Proceedings of thesixth Asian logic conference, World Scientific Publishing Co. Inc., River Edge, 1998, pp. 121–149.[KLP10] David Kerr, Hanfeng Li, and Mika¨el Pichot, Turbulence, representations, and trace-preserving actions ,Proceedings of the London Mathematical Society (2010), no. 2, 459–484.[KM16] Alexander S. Kechris and Henry L. Macdonald, Borel equivalence relations and cardinal algebras , Fun-damenta Mathematicae (2016), no. 2, 183–198.[KS77] Jerome Kaminker and Claude Schochet, K -theory and Steenrod homology: applications to the Brown-Douglas-Fillmore theory of operator algebras , Transactions of the American Mathematical Society (1977), 63–107.[KS17] Jerome Kaminker and Claude L. Schochet, Spanier–Whitehead K-duality for C*-algebras , Journal ofTopology and Analysis (2017), no. 01, 21–52, Publisher: World Scientific Publishing Co.[Lan95] E. Christopher Lance, Hilbert C*-modules , London Mathematical Society Lecture Note Series, vol. 210,Cambridge University Press, Cambridge, 1995.[Lup14] Martino Lupini, Unitary equivalence of automorphisms of separable C*-algebras , Advances in Mathe-matics (2014), 1002–1034.[Lup20] , Definable Eilenberg–Mac Lane Universal Coefficient Theorems , arXiv:2009.10805 (2020).[Mac57] George W. Mackey, Borel structure in groups and their duals , Transactions of the American Mathemat-ical Society (1957), no. 1, 134–165.[Mil62] John Milnor, On axiomatic homology theory , Pacific Journal of Mathematics (1962), 337–341.[Mil95] , On the Steenrod homology theory , Novikov conjectures, index theorems and rigidity, Vol. 1(Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cam-bridge, 1995, pp. 79–96.[ML98] Saunders Mac Lane, Categories for the working mathematician , second ed., Graduate Texts in Mathe-matics, vol. 5, Springer-Verlag, New York, 1998.[MR12] Luca Motto Ros, On the complexity of the relations of isomorphism and bi-embeddability , Proceedingsof the American Mathematical Society (2012), no. 1, 309–323.[MS82] Sibe Mardeˇsi´c and Jack Segal, Shape theory , North-Holland Mathematical Library, vol. 26, North-Holland Publishing Co., Amsterdam-New York, 1982.[Mur90] Gerard J. Murphy, C*-algebras and operator theory , Academic Press, Inc., Boston, MA, 1990.[NdK17] Ryszard Nest and Niek de Kleijn, Excision and Bott periodicity , 2017, Available at http://niekdekleijn.com/wp-content/uploads/2017/11/excision_BP.pdf .[Pas81] William L. Paschke, K -theory for commutants in the Calkin algebra , Pacific Journal of Mathematics (1981), no. 2, 427–434.[Pau02] Vern I. Paulsen, Completely bounded maps and operator algebras , Cambridge Studies in AdvancedMathematics, vol. 78, Cambridge University Press, Cambridge, 2002.[Ped79] Gert K. Pedersen, C*-algebras and their automorphism groups , London Mathematical Society Mono-graphs, vol. 14, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1979.[Ped89] , Analysis now , Graduate Texts in Mathematics, vol. 118, Springer-Verlag, New York, 1989.[RLL00] Mikael Rørdam, Flemming Larsen, and Niels Laustsen, An introduction to K-theory for C*-algebras ,London Mathematical Society Student Texts, vol. 49, Cambridge University Press, Cambridge, 2000.[Rør95] Mikael Rørdam, Classification of certain infinite simple C*-algebras , Journal of Functional Analysis (1995), no. 2, 415–458.[Rot09] Joseph J. Rotman, An introduction to homological algebra , Universitext, Springer, New York, 2009.[RR64] A. P. Robertson and W. J. Robertson, Topological vector spaces , Cambridge Tracts in Mathematics andMathematical Physics, No. 53, Cambridge University Press, New York, 1964. EFINABLE K-HOMOLOGY OF SEPARABLE C*-ALGEBRAS 63 [RS81] Jonathan Rosenberg and Claude Schochet, Comparing functors classifying extensions of C*-algebras ,Journal of Operator Theory (1981), no. 2, 267–282.[RS87] , The K¨unneth Theorem and the Universal Coefficient Theorem for Kasparov’s generalized K -functor , Duke Mathematical Journal (1987), no. 2, 431–474.[RS02] Mikael Rørdam and Erling Størmer, Classification of nuclear C*-algebras. Entropy in operator algebras ,Encyclopaedia of Mathematical Sciences, vol. 126, Springer-Verlag, Berlin, 2002.[Rue06] David Ruelle, What is . . . a strange attractor? , Notices of the American Mathematical Society (2006), no. 7, 764–765.[RW98] Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras , MathematicalSurveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998.[Sal92] Norberto Salinas, Relative quasidiagonality and KK -theory , Houston Journal of Mathematics (1992),no. 1, 97–116.[Sch84] Claude Schochet, Topological methods for C*-algebras. III. Axiomatic homology , Pacific Journal of Math-ematics (1984), no. 2, 399–445.[Sch96] Claude L. Schochet, The UCT, the Milnor sequence, and a canonical decomposition of the Kasparovgroups , K-Theory (1996), no. 1, 49–72.[Sch01] , The fine structure of the Kasparov groups. I. Continuity of the KK -pairing , Journal of Func-tional Analysis (2001), no. 1, 25–61.[Sch02] , The fine structure of the Kasparov groups. II. Topologizing the UCT , Journal of FunctionalAnalysis (2002), no. 2, 263–287.[Sch03] , A Pext primer: pure extensions and lim for infinite abelian groups , New York Journal ofMathematics. NYJM Monographs, vol. 1, State University of New York, University at Albany, Albany,NY, 2003.[Sch05] , The fine structure of the Kasparov groups. III. Relative quasidiagonality , Journal of OperatorTheory (2005), no. 1, 91–117.[Sim95] Barry Simon, Operators with singular continuous spectrum. I. general operators , Annals of Mathematics.Second Series (1995), no. 1, 131–145.[Sma67] Stephen Smale, Differentiable dynamical systems , Bulletin of the American Mathematical Society (1967), 747–817.[Tho15] Simon Thomas, A descriptive view of unitary group representations , Journal of the European Mathe-matical Society (JEMS) (2015), no. 7, 1761–1787.[Val85] Jean-Michel Vallin, C*-alg`ebres de Hopf et C *-alg`ebres de Kac , Proceedings of the London MathematicalSociety. Third Series (1985), no. 1, 131–174.[vD32] David van Dantzig, Theorie des projektiven Zusammenhangs n -dimensionaler R¨aume , MathematischeAnnalen (1932), no. 1, 400–454.[Vie27] Leopold Vietoris, ¨Uber den h¨oheren Zusammenhang kompakter R¨aume und eine Klasse von zusammen-hangstreuen Abbildungen , Mathematische Annalen (1927), no. 1, 454–472.[Voi76] Dan Voiculescu, A non-commutative Weyl-von Neumann theorem , Revue Roumaine de Math´ematiquesPures et Appliqu´ees (1976), no. 1, 97–113.[Wil74] Robert F. Williams, Expanding attractors , Institut des Hautes ´Etudes Scientifiques. PublicationsMath´ematiques (1974), no. 43, 169–203.[WN92] Stanis law L. Woronowicz and Kazimierz Napi´orkowski, Operator theory in the C*-algebra framework ,Reports on Mathematical Physics (1992), no. 3, 353–371.[WO93] N. E. Wegge-Olsen, K -theory and C*-algebras , Oxford Science Publications, The Clarendon Press,Oxford University Press, New York, 1993.[Wor80] Stanis law L. Woronowicz, Pseudospaces, pseudogroups and Pontriagin duality , Mathematical problemsin theoretical physics (Proc. Internat. Conf. Math. Phys., Lausanne, 1979), Lecture Notes in Phys., vol.116, Springer, Berlin-New York, 1980, pp. 407–412.[Wor91] , Unbounded elements affiliated with C*-algebras and noncompact quantum groups , Communica-tions in Mathematical Physics (1991), no. 2, 399–432.[Wor95] , C*-algebras generated by unbounded elements , Reviews in Mathematical Physics (1995), no. 3,481–521. [Zek89] Richard Zekri, A new description of Kasparov’s theory of C*-algebra extensions , Journal of FunctionalAnalysis (1989), no. 2, 441–471. School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, 6140 Wellington, New Zealand Email address ::