Turbulence, orbit equivalence, and the classification of nuclear C*-algebras
TTURBULENCE, ORBIT EQUIVALENCE, AND THECLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS ILIJAS FARAH, ANDREW S. TOMS AND ASGER T ¨ORNQUIST
Abstract.
We bound the Borel cardinality of the isomorphism relation for nuclear simple separableC ∗ -algebras: It is turbulent, yet Borel reducible to the action of the automorphism group of theCuntz algebra O on its closed subsets. The same bounds are obtained for affine homeomorphismof metrizable Choquet simplexes. As a by-product we recover a result of Kechris and Solecki,namely, that homeomorphism of compacta in the Hilbert cube is Borel reducible to a Polish groupaction. These results depend intimately on the classification theory of nuclear simple C ∗ -algebrasby K-theory and traces. Both of necessity and in order to lay the groundwork for further studyon the Borel complexity of C ∗ -algebras, we prove that many standard C ∗ -algebra constructionsand relations are Borel, and we prove Borel versions of Kirchberg’s O -stability and embeddingtheorems. We also find a C ∗ -algebraic witness for a K σ hard equivalence relation. The authors dedicate this article to the memory of Greg Hjorth. Introduction
The problem of classifying a category of objects by assigning objects of another category ascomplete invariants is fundamental to many disciplines of mathematics. This is particularly truein C ∗ -algebra theory, where the problem of classifying the nuclear simple separable C ∗ -algebras upto isomorphism is a major theme of the modern theory. Recent contact between descriptive settheorists and operator algebraists has highlighted two quite different views of what it means to havesuch a classification. Operator algebraists have concentrated on finding complete invariants whichare assigned in a functorial manner, and for which there are good computational tools (K-theory,for instance.) Descriptive set theorists, on the other hand, have developed an abstract degree theory of classification problems, and have found tools that allow us to compare the complexity of differentclassification problems, and, importantly, allow us to rule out the use of certain types of invariantsin a complete classification of highly complex concrete classification problems.The aim of this paper is to investigate the complexity of the classification problem for nuclearsimple separable C ∗ -algebras from the descriptive set theoretic point of view. A minimal require-ment of any reasonable classification is that the invariants are somehow definable or calculablefrom the objects being classified themselves. For example, it is easily seen that there are at mostcontinuum many non-isomorphic separable C ∗ -algebras, and so it is possible, in principle, to assignto each isomorphism class of separable C ∗ -algebras a unique real number, thereby classifying theseparable C ∗ -algebras completely up to isomorphism. Few mathematicians working in C ∗ -algebraswould find this a satisfactory solution to the classification problem for separable C ∗ -algebras, letalone nuclear simple separable C ∗ -algebras, since we do not obtain a way of computing the invariant,and therefore do not have a way of effectively distinguishing the isomorphism classes. Date : November 10, 2018. a r X i v : . [ m a t h . OA ] A p r ILIJAS FARAH, ANDREW S. TOMS AND ASGER T ¨ORNQUIST
Since descriptive set theory is the theory of definable sets and functions in Polish spaces, itprovides a natural framework for a theory of classification problems. In the past 30 years, suchan abstract theory has been developed. This theory builds on the fundamental observation thatin most cases where the objects to be classified are themselves either countable or separable, thereis a natural standard Borel space which parameterizes (up to isomorphism) all the objects in theclass. From a descriptive set theoretic point of view, a classification problem is therefore a pair(
X, E ) consisting of a standard Borel space X , the (parameters for) objects to be classified, and anequivalence relation E , the relation of isomorphism among the objects in X . In most interestingcases, the equivalence relation E is easily definable from the elements of X , and is seen to be Borelor, at worst, analytic. Definition 1.1.
Let (
X, E ) and (
Y, F ) be classification problems, in the above sense. A
Borelreduction of E to F is a Borel function f : X → Y such that xEy ⇐⇒ f ( x ) F f ( y ) . If such a function f exists then we say that E is Borel reducible to F , and we write E ≤ B F .If f is a Borel reduction of E to F , then evidently f provides a complete classification of thepoints of X up to E equivalence by an assignment of F equivalence classes. The “effective”descriptive set theory developed in 1960s and 1970s (see e.g. [20]) established in a precise waythat the class of Borel functions may be thought of as a very general class of calculable functions.Therefore the notion of Borel reducibility provides a natural starting point for a systematic theoryof classification which is both generally applicable, and manages to ban the trivialities provided bythe Axiom of Choice. Borel reductions in operator algebras have been studied in the recent workof Sasyk-T¨ornquist [26, 25, 27], who consider the complexity of isomorphism for various classes ofvon Neumann factors, and in that of Kerr-Li-Pichot [16] and Farah [5], who concentrate on certainrepresentation spaces and, in [16], group actions on the hyperfinite II factor. This article initiatesthe study of Borel reducibility in separable C ∗ -algebras.In [14], Kechris introduced a standard Borel structure on the space of separable C ∗ -algebras,providing a natural setting for the study of the isomorphism relation on such algebras. Thisrelation is of particular interest for the subset of (unital) nuclear simple separable C ∗ -algebras, asthese are the focus of G. A. Elliott’s long running program to classify such algebras via K-theoreticinvariants. To situate our main result for functional analysts, let us mention that an attractiveclass of invariants to use in a complete classification are the countable structures type invariants,which include the countable groups and countable ordered groups, as well as countable graphs,fields, boolean algebras, etc. If ( X, E ) is a classification problem, we will say that E is classifiableby countable structures if there is a Borel reduction of E to the isomorphism relation for somecountable structures type invariant. If ( X, E ) is not classifiable by countable structures, then itmay still allow some reasonable classification, in the sense that it is Borel reducible to the orbitequivalence relation of a Polish group action on a standard Borel space. Our main result is thefollowing theorem (which is proved in § § Theorem 1.2.
The isomorphism relation E for unital simple separable nuclear C ∗ -algebras isturbulent, hence not classifiable by countable structures. Moreover, if L is any countable languageand (cid:39) Mod( L ) denotes the isomorphism relation for countable models of L , then (cid:39) Mod( L ) is Borelreducible to E . On the other hand, E is Borel reducible to the orbit equivalence relation of a Polishgroup action, namely, the action of Aut( O ) on the closed subsets of O . URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 3 This establishes that the isomorphism problem for nuclear simple separable unital C ∗ -algebrasdoes not have the maximal complexity among analytic classification problems, and rules out theusefulness of some additional types of invariants for a complete classification of nuclear simpleseparable unital C ∗ -algebras. It also establishes that this relation has higher complexity than theisomorphism relation of any class of countable structures. Remarkably, establishing both the lowerand upper ≤ B bounds of Theorem 1.2 requires that we prove Borel versions of two well-knownresults from Elliott’s K-theoretic classification program for nuclear simple separable C ∗ -algebras.The lower bound uses the classification of the unital simple approximately interval (AI) algebrasvia their K -group and simplex of tracial states, while the upper bound requires that we prove aBorel version of Kirchberg’s Theorem that a simple unital nuclear separable C ∗ -algebra satisfies A ⊗ O ∼ = O .By contrast with Theorem 1.2, we shall establish in [6] that Elliott’s classification of unital AFalgebras via the ordered K -group amounts to a classification by countable structures. This willfollow from a more general result regarding the Borel computability of the Elliott invariant. Wenote that there are non-classification results in the study of simple nuclear C ∗ -algebras which ruleout the possibility of classifying all simple nuclear separable C ∗ -algebras via the Elliott invariantin a functorial manner (see [23] and [29]). At heart, these examples exploit the structure of theCuntz semigroup, an invariant whose descriptive set theory will be examined in [6].The proof of Theorem 1.2 allows us to draw conclusions about the complexity of metrizableChoquet simplexes, too. Theorem 1.3.
The relation of affine homeomorphism on metrizable Choquet simplexes is turbulent,yet Borel reducible to the orbit equivalence relation of a Polish group action.
Furthermore, and again as a by-product of Theorem 1.2, we recover an unpublished result of Kechrisand Solecki:
Theorem 1.4 (Kechris-Solecki, 2006) . The relation of homeomorphism on compact subsets of theHilbert cube is Borel reducible to the orbit equivalence relation of a Polish group action.
Finally, we show that a Borel equivalence relation which is not
Borel reducible to any orbitequivalence relation of a Polish group action has a C*-algebraic witness. Recall that E K σ is thecomplete K σ equivalence relation. Theorem 1.5. E K σ is Borel reducible to bi-embeddability of unital AF algebras. The early sections of this paper are dedicated to establishing that a variety of standard con-structions in C ∗ -algebra theory are Borel computable, and that a number of important theorems inC ∗ -algebra theory have Borel computable counterparts. This is done both of necessity—Theorems1.2–1.5 depend on these facts—and to provide the foundations for a general theory of calculabilityfor constructions in C ∗ -algebra theory. Constructions that are shown to be Borel computable in-clude passage to direct limits, minimal tensor products, unitization, and the calculation of states,pure states, and traces. Theorems for which we establish Borel counterparts include Kirchberg’sExact Embedding Theorem, as well as Kirchberg’s A ⊗O (cid:39) O Theorem for unital simple separablenuclear A .Figure 1 summarizes the Borel reductions we obtain in this article, in addition to some knownreductions. All classes of C*-algebras occurring in the diagram are unital and separable. Unlessotherwise specified, the equivalence relation on a given class is the isomorphism relation. The bi-reducibility between the isomorphism for UHF algebras and bi-embeddability of UHF algebras is ILIJAS FARAH, ANDREW S. TOMS AND ASGER T ¨ORNQUIST isomorphism ofBanach spacesbiembeddabilityof AF algebras Cuntzsemigroups E K σ E X ∞ G ∞ simple nuclear C*-algebras Elliott invariantssimple AI algebrasChoquet simplexeshomeomorphism ofcompact metric spaces abelian C*-algebras E AF algebras E UHF biembeddabilityof UHF algebras orbit equivalence relationscountable structuressmooth
Figure 1.
Borel reducibility diagraman immediate consequence of Glimm’s characterization of UHF algebras, or rather of its (straight-forward) Borel version. E denotes the eventual equality relation in the space 2 N . The fact that E is the minimal non-smooth Borel equivalence relation is the Glimm–Effros dichotomy, provedby Harrington, Kechris and Louveau (see [10]). E denotes the eventual equality relation in [0 , N .By [15] E is not Borel-reducible to any Polish group action. E K σ is the complete K σ equivalencerelation, and E X ∞ G ∞ is the maximal orbit equivalence relation of a Polish group action (see [10]). Thenontrivial direction of Borel bi-reducibility between abelian C*-algebras and compact metric spaces URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 5 follows from Lemma 3.17. A Borel reduction from compact metric spaces to Choquet simplexesis given in Lemma 4.8. A Borel reduction from Choquet simplexes to simple AI algebras is givenin Corollary 5.2. The Borel version of Elliott’s reduction of simple AI algebras to Elliott invariantfollows from Elliott’s classification result and the fact that the computation of the Elliott invariantis Borel, proved in [6]. The reduction of the Elliott invariant to E X ∞ G ∞ , as well as the facts aboutthe Cuntz semigroup, is proved in [6]. Bi-embeddability of AF algebras is proved to be above E K σ in Section 8. The isomorphism of separable Banach spaces is the complete analytic equivalencerelation by [30]. Some of the reductions in Figure 1 are not known to be sharp. For example, it isnot known whether the homeomorphism of compact metric spaces is equireducible with E X ∞ G ∞ (cf.remark at the end of § ∗ -algebras. In section 3 we prove that most standard constructions in C ∗ -algebratheory correspond to Borel functions and relations. In Section 4 we give a parameterization ofthe set of metrizable separable Choquet simplexes. In section 5 we establish the lower bound ofTheorem 1.2. A Borel version of Kirchberg’s Exact Embedding Theorem is obtained in section 6.The upper bound in Therorem 1.2 is proved in Section 7 using the Borel version of Kirchberg’s A ⊗ O (cid:39) O Theorem; Theorem 1.3 is also established. Section 8 establishes Theorem 1.5, andSection 9 discusses several questions that remain open and warrant further investigation.
Acknowledgements.
Ilijas Farah was partially supported by NSERC. A. Toms was supportedby NSF grant DMS-0969246 and the 2011 AMS Centennial Fellowship. Asger T¨ornquist wishes toacknowledge generous support from the following grants: The Austrian Science Fund FWF grantno. P 19375-N18, The Danish Council for Independent Research (Natural Sciences) grant no. 10-082689/FNU, and Marie Curie re-integration grant no. IRG- 249167, from the European Union.We would like to thank the referee for a very useful report.2.
Parameterizing separable C ∗ -algebras In this section we describe several standard Borel spaces that in a natural way parameterize theset of all separable C ∗ -algebras. To make this precise, we adopt the following definition. Definition 2.1.
Let C be a category of objects.(1) A standard Borel parameterization of C is a pair ( X, f ) consisting of standard Borel space X and a function f : X → C such that f ( X ) meets each isomorphism class in C . (For brevity,we often simply call ( X, f ) a parameterization of C . We will also usually abuse notation bysuppressing f and writing X . Finally, note that despite the terminology, it is X rather thanthe parameterization that is standard Borel.)(2) The equivalence relation (cid:39) ( X,f ) on X is defined by x (cid:39) ( X,f ) y ⇐⇒ f ( x ) is isomorphic to f ( y ) . (3) A parameterization ( X, f ) is called good if (cid:39) ( X,f ) is analytic as a subset of X × X .(4) Let ( X, f ) and (
Y, g ) be two parameterizations of the same category C . A homomorphism of ( X, f ) to (
Y, g ) is a function ψ : X → Y such that for ψ ( x ) is isomorphic to g ( ψ ( x )) for all ILIJAS FARAH, ANDREW S. TOMS AND ASGER T ¨ORNQUIST x ∈ X . An isomorphism of ( X, f ) and (
Y, g ) is a bijective homomorphism; a monomorphismis an injective homomorphism.(5) We say that (
X, f ) and (
Y, g ) are equivalent if there is a Borel isomorphism from (
X, f ) to(
Y, g ).(6) We say that (
X, f ) and (
Y, g ) are weakly equivalent if there are Borel homomorphisms ψ : X → Y of ( X, f ) to (
Y, g ) and φ : Y → X of ( Y, g ) to (
X, f ).When f is clear from the context, we will allow a slight abus de langage and say that X is aparameterization of C when ( X, f ) is. Further, we will usually write (cid:39) X for (cid:39) ( X,f ) . Note that bythe Borel Schr¨oder-Bernstein Theorem ([13, Theorem 15.7]), (5) is equivalent to(5’) There are Borel monomorphisms ψ : X → Y of ( X, f ) to (
Y, g ) and φ : Y → X of ( Y, g ) to(
X, f ).We now introduce four different parameterizations of the class of separable C ∗ -algebras, whichwe will later see are all (essentially) equivalent and good.2.1. The space Γ( H ) . Let H be a separable infinite dimensional Hilbert space and let as usual B ( H ) denote the space of bounded operators on H . The space B ( H ) becomes a standard Borelspace when equipped with the Borel structure generated by the weakly open subsets. Following[14] we let Γ( H ) = B ( H ) N , and equip this with the product Borel structure. Every γ ∈ Γ( H ) is identified with a sequence γ n ,for n ∈ N , of elements of B ( H ). For each γ ∈ Γ( H ) we let C ∗ ( γ ) be the C ∗ -algebra generated bythe sequence γ . If we identify each γ ∈ Γ( H ) with the C ∗ ( γ ), then naturally Γ( H ) parameterizesall separable C ∗ -algebras acting on H . Since every separable C ∗ -algebra is isomorphic to a C ∗ -subalgebra of B ( H ) this gives us a standard Borel parameterization of the category of all separableC ∗ -algebras. If the Hilbert space H is clear from the context we will write Γ instead of Γ( H ).Following Definition 2.1, we define γ (cid:39) Γ γ (cid:48) ⇐⇒ C ∗ ( γ ) is isomorphic to C ∗ ( γ (cid:48) ) . The space ˆΓ( H ) . Let Q ( i ) = Q + i Q denote the complex rationals. Following [14], let( p j : j ∈ N ) enumerate the non-commutative ∗ -polynomials without constant term in the formalvariables X k , k ∈ N , with coefficients in Q ( i ), and for γ ∈ Γ write p j ( γ ) for the evaluation of p j with X k = γ ( k ). Then C ∗ ( γ ) is the norm-closure of { p j ( γ ) : j ∈ N } . The map Γ → Γ : γ (cid:55)→ ˆ γ where ˆ γ ( j ) = p j ( γ ) is clearly a Borel map from Γ to Γ. If we letˆΓ( H ) = { ˆ γ : γ ∈ Γ( H ) } , then ˆΓ( H ) is a standard Borel space and provides another parameterization of the C ∗ -algebrasacting on H ; we suppress H and write ˆΓ whenever possible. For γ ∈ ˆΓ, let ˇ γ ∈ Γ be defined byˇ γ ( n ) = γ ( i ) ⇐⇒ p i = X n , and note that ˆΓ → Γ : γ (cid:55)→ ˇ γ is the inverse of Γ → ˆΓ : γ (cid:55)→ ˆ γ . We let (cid:39) ˆΓ be γ (cid:39) ˆΓ γ (cid:48) ⇐⇒ C ∗ ( γ ) is isomorphic to C ∗ ( γ (cid:48) ) . It is clear from the above that Γ and ˆΓ are equivalent parameterizations.
URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 7 An alternative picture of ˆΓ( H ) is obtained by considering the free (i.e., surjectively universal)countable unnormed Q ( i )- ∗ -algebra A . We can identify A with the set { p n : n ∈ N } . ThenˆΓ A ( H ) = { f : A → B ( H ) : f is a ∗ -homomorphism } is easily seen to be a Borel subset of B ( H ) A . For f ∈ ˆΓ A let C ∗ ( f ) be the norm closure of im( f ),and define f (cid:39) ˆΓ A f (cid:48) ⇐⇒ C ∗ ( f ) is isomorphic to C ∗ ( f (cid:48) ) . Clearly the map ˆΓ → ˆΓ A : γ (cid:55)→ f γ defined by f γ ( p j ) = γ ( j ) provides a Borel bijection witnessingthat ˆΓ and ˆΓ A are equivalent (and therefore they are also equivalent to Γ.)We note for future reference that if we instead consider the free countable unital unnormed Q ( i )- ∗ -algebra A u and letˆΓ A u ( H ) = { f : A u → B ( H ) : f is a unital ∗ -homomorphism } , then this gives a parameterization of all unital C ∗ -subalgebras of B ( H ). Note that A u may beidentified with the set of all formal ∗ -polynomials in the variables X k with coefficients in Q ( i )(allowing a constant term.)2.3. The space Ξ . Consider the Polish space R N . We let Ξ be the space of all δ ∈ R N such thatfor some separable C ∗ -algebra A and a sequence y = ( y n ) in A generating it we have that δ ( j ) = (cid:107) p j ( y ) (cid:107) A . Each δ ∈ Ξ defines a seminorm (cid:107) p j (cid:107) δ = δ ( j ) on A which satisfies the C ∗ -axiom. Letting I = { p j : δ ( j ) = 0 } we obtain a norm on A /I . The completion of this algebra is then a C ∗ -algebra, whichwe denote by B ( δ ). It is clearly isomorphic to any C ∗ -algebra A with y = ( y n ) as above satisfying (cid:107) p j ( y ) (cid:107) = δ ( j ). Lemma 2.2.
The set Ξ is closed in R N .Proof. Assume δ n ∈ Ξ converges to δ ∈ R N pointwise. Fix C ∗ -algebras A n and sequences y n =( y ni ∈ A n : i ∈ N ) such that δ n ( j ) = (cid:107) p j ( y n ) (cid:107) A n for all n and j . For a nonprincipal ultrafilter U on N , let A ∞ be the subalgebra of the ultraproduct (cid:81) U A n generated by the elements y i = ( y i , y i , . . . ),for i ∈ N . Then clearly δ ( j ) = lim n →U (cid:107) p j ( y n ) (cid:107) A n = (cid:107) p j ( y i ) (cid:107) A ∞ , hence A witnesses δ ∈ Ξ. (cid:3) Thus Ξ provides yet another parameterization of the category of separable C ∗ -algebras, and wedefine in Ξ the equivalence relation δ (cid:39) Ξ δ (cid:48) ⇐⇒ B ( δ ) is isomorphic to B ( δ (cid:48) ) . Below we will prove that this parameterization is equivalent to Γ and ˆΓ. Note that an alternativedescription of Ξ is obtained by considering the set of f ∈ R A which define a C ∗ -seminorm on A ;this set is easily seen to be Borel since the requirements of being C ∗ -seminorm are Borel conditions.2.4. The space ˆΞ . Our last parameterization is obtained by considering the setˆΞ ⊆ N N × N × N Q ( i ) × N × N N × N × N N × R N ILIJAS FARAH, ANDREW S. TOMS AND ASGER T ¨ORNQUIST of all tuples ( f, g, h, k, r ) such that the operations (with m, n in N and q ∈ Q ( i )) defined by m + f n = f ( m, n ) q · g n = g ( q, n ) m · h n = h ( m, n ) m ∗ k = k ( m ) (cid:107) n (cid:107) r = r ( n )give N the structure of a normed ∗ -algebra over Q ( i ) which further satisfies the “C ∗ -axiom”, (cid:107) n · h n ∗ k (cid:107) r = (cid:107) n (cid:107) r for all n ∈ N . The set ˆΞ is Borel since the axioms of being a normed ∗ -algebra over Q ( i ) are Borelconditions. For A ∈ ˆΞ, let ˆ B ( A ) denote the completion of A with respect to the norm and equippedwith the extension of the operations on A to ˆ B ( A ). Note in particular that the operation of scalarmultiplication may be uniquely extended from Q ( i ) to C . We define for A , A ∈ ˆΞ the equivalencerelation A (cid:39) ˆΞ A ⇐⇒ ˆ B ( A ) is isomorphic to ˆ B ( A ) . For future reference, we note that the infinite symmetric group Sym( N ) acts naturally on ˆΞ: If σ ∈ Sym( N ) and ( f, g, h, k, r ) ∈ ˆΞ, we let σ · f ∈ N N × N be defined by( σ · f )( m, n ) = k ⇐⇒ f ( σ − ( m ) , σ − ( n )) = σ − ( k ) , and defined σ · g , σ · h , σ · k and σ · r similarly. Then we let σ · ( f, g, h, k, r ) = ( σ · f, σ · g, σ · h, σ · k, σ · r ). Itis clear that σ induces an isomorphism of the structures ( f, g, h, k, r ) and σ · ( f, g, h, k, r ). However,it clearly does not induce the equivalence relation (cid:39) ˆΞ , which is strictly coarser. Remark . (1) It is useful to think of Γ and ˆΓ as parameterizations of concrete C ∗ -algebras, whileΞ and ˆΞ can be thought of as parameterizing abstract C ∗ algebras.(2) The parameterizations Γ, ˆΓ and Ξ all contain a unique element corresponding to the trivialC ∗ -algebra, which we denote by 0 in all cases. Note that ˆΞ does not parameterize the trivialC ∗ -algebra.2.5. Equivalence of Γ , ˆΓ , Ξ and ˆΞ . We now establish that the four parameterizations describedabove give us equivalent parameterizations of the non-trivial separable C ∗ -algebras. First we needthe following lemma. Lemma 2.4.
Let X be a Polish space and let Y be any of the spaces Γ , ˆΓ , Ξ or ˆΞ . Let f : X → Y be a Borel function such that f ( x ) (cid:54) = 0 for all x ∈ X . Then there is a Borel injection ˜ f : X → Y such that for all x ∈ X , f ( x ) (cid:39) Y ˜ f ( x ) .Proof. Y = Γ: We may assume that X = [ N ] ∞ , the space of infinite subsets of N . (Under thenatural identification this is a G δ subset of 2 N and therefore Polish. It is then homeomorphic tothe set of irrationals.) Given γ ∈ Γ \ { } and x ∈ X , let n ( γ ) ∈ N be the least such that γ ( n ) (cid:54) = 0and define γ x ( k ) = iγ ( n ( γ )) if k = 2 i for some i ∈ x ; γ ( j ) if k = 3 j for some j ∈ N ;0 otherwise . Clearly C ∗ ( γ ) = C ∗ ( γ x ), and ˜ f : X → Γ \ { } defined by ˜ f ( x ) = ( f ( x )) x is a Borel injection. URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 9 Y = ˆΓ: Clear, since Γ and ˆΓ are equivalent. Y = Ξ. We may assume that X = R + . Fix x ∈ X , let δ = f ( x ), and let n ( δ ) ∈ N be least suchthat p n = X i for some i ∈ N and δ ( n ) (cid:54) = 0. Let A be a C ∗ -algebra and y = ( y n ) be a densesequence in A such that δ ( n ) = (cid:107) p n ( y ) (cid:107) A , and let ˜ y = (˜ y n ) be˜ y n = (cid:26) x (cid:107) y n (cid:107) A y n if n = i ; y n otherwise.Then define ˜ f ( x )( n ) = (cid:107) p n (˜ y ) (cid:107) A . Clearly ˜ f ( x ) (cid:39) Ξ f ( x ) for all x ∈ X , and since (cid:107) p n (˜ y ) (cid:107) A = x ,the function ˜ f is injective. (Note that ˜ f ( x ) does not depend on the choice of A and y , so it is infact a function.) Finally, ˜ f is Borel by [13, 14.12], since˜ f ( x ) = δ (cid:48) ⇐⇒ ( ∃ γ, γ (cid:48) ∈ Γ)( ∃ δ ∈ Ξ \ { } ) f ( x ) = δ ∧ ( ∀ n ) δ ( n ) = (cid:107) p n ( γ ) (cid:107)∧ ( ∀ i )(( i (cid:54) = n ( δ ) ∧ γ (cid:48) ( i ) = γ ( i )) ∨ ( i = n ( δ ) ∧ γ (cid:48) ( i ) = x (cid:107) γ ( i ) (cid:107) γ ( i ))) , gives an analytic definition of the graph of ˜ f (with n ( δ ) defined as above.) Y = ˆΞ: Assume X = [2 N ] ∞ , the infinite subsets of the even natural numbers. Given ( f, g, h, k, r ) ∈ ˆΞ and x ∈ X , we can find, in a Borel way, a permutation σ = σ ( f,g,h,j,r ) ,x of N so that x = { n · σ · g n ∈ N } . Then ˜ f ( x ) = σ ( f,g,h,j,r ) ,x · f ( x ) works. (cid:3) Remark . The classical principle [13, 14.12] that a function whose graph is analytic is Borel willbe used frequently in what follows, usually without comment.
Proposition 2.6. Ξ \ { } and ˆΞ are equivalent.Proof. By the previous Lemma, it suffices to show that Ξ \ { } and ˆΞ are weakly equivalent.Identify Ξ \ { } with a subset of R A in the natural way, and define E = { ( δ, m, n ) ∈ Ξ \ { } × N × N : δ ( p m − p n ) = 0 } . Then the section E δ = { ( m, n ) ∈ N : ( δ, m, n ) ∈ E } defines an equivalence relation on N . Let f n : Ξ → N be Borel functions such that each f n ( δ ) is the least element in N not E δ -equivalent to f m ( δ ) for m < n . If we let I δ = { p n : δ ( p n ) = 0 } , then n (cid:55)→ f ( n ) I δ provides a bijection between A /I δ and N , and from this we can define (in a Borel way) algebra operations and the norm on N corresponding to A ∈ ˆΞ such that A (cid:39) A /I δ .Conversely, given a normed Q ( i )- ∗ -algebra A ∈ ˆΞ (with underlying set N ), an element δ A ∈ Ξ isdefined by letting δ A ( n ) = (cid:107) p n ( X i = i : i ∈ N ) (cid:107) A , where p n ( X i = i : i ∈ N ) denotes the evaluationof p n in A when letting X i = i . (cid:3) Proposition 2.7. Γ and Ξ are equivalent. Thus Γ , ˆΓ and Ξ are equivalent parameterizations ofthe separable C ∗ -algebras, and Γ \ { } , ˆΓ \ { } , Ξ \ { } and ˆΞ are equivalent parameterizations ofthe non-trivial separable C ∗ -algebras. For the proof of this we need the following easy (but useful) Lemma:
Lemma 2.8.
Let H be a separable infinite dimensional Hilbert space. Then:(1) A function f : X → Γ( H ) on a Polish space X is Borel if and only if for some (any) sequence ( e i ) with dense span in H we have that the functions x (cid:55)→ ( f ( x )( n ) e i | e j ) are Borel, for all n, i, j ∈ N .(2) Suppose g : X → (cid:83) x ∈ X Γ( H x ) is a function such that for each x ∈ X we have g ( x ) ∈ Γ( H x ) ,where H x is a separable infinite dimensional Hilbert space, and there is a system ( e xi ) j ∈ N with span { e xi : i ∈ N } dense in H x . If for all n, i, j ∈ N we have that the functions X → C : x (cid:55)→ ( e xi | e xj ) and X → C : x (cid:55)→ ( g ( n )( x ) e xi | e xj ) are Borel, then there is a Borel ˆ g : X → B ( H ) and a family T x : H → H x of linear isometries suchthat for all n ∈ N , g ( x )( n ) = T x ˆ g ( x )( n ) T − x . We postpone the proof of Lemma 2.8 until after the proof of Proposition 2.7.
Proof of Proposition 2.7.
By Lemma 2.4, it is again enough to show that Γ and Ξ are weaklyequivalent. For the first direction, the map ψ : Γ → Ξ given by ψ ( γ )( n ) = (cid:107) p n ( γ ) (cid:107) clearly works.For the other direction we rely on the GNS construction (e.g. [2, II.6.4]). For each δ ∈ Ξ let S ( δ ) be the space of all φ ∈ C N such that(1) | φ ( k ) | ≤ δ ( k ) for all k ,(2) φ ( k ) = φ ( m ) + φ ( n ), whenever p k = p m + p n ,(3) φ ( k ) ≥ p k = p ∗ m p m for some m .Then S ( δ ) is a compact subset of C N for each δ ∈ Ξ, and so since the relation { ( δ, φ ) ∈ Ξ × C N : φ ∈ S ( δ ) } is Borel, it follows by [13, 28.8] that Ξ → K ( C N ) : δ (cid:55)→ S ( δ ) is a Borel function into the Polishspace K ( C N ) of compact subsets of C N . Consider the set N = { ( δ, φ, n, m ) ∈ Ξ × C N × N × N : φ ∈ S ( δ ) ∧ ( ∃ k ) p k = ( p n − p m ) ∗ ( p n − p m ) ∧ φ ( k ) = 0 } . Then for each δ and φ the relation N δ,φ = { ( n, m ) ∈ N : ( δ, φ, n, m ) ∈ N } is an equivalencerelation on N . Without any real loss of generality we can assume that N δ,φ always has infinitelymany classes. Let σ n : Ξ × C N → N be a sequence of Borel maps such that for all δ and φ fixed theset { σ n ( δ, φ ) : n ∈ N } meets every N δ,φ class once. For δ and φ fixed we can then define an inner product on N by( n | m ) δ,φ = φ ( k ) ⇐⇒ p k = p ∗ σ n ( δ,φ ) p σ m ( δ,φ ) . Let H ( δ, φ ) denote the completion of this pre-Hilbert space. Then there is a unique operator γ ( n ) ∈ B ( H ( δ, φ )) extending the operator acting on ( N , ( ·|· ) δ,φ ) defined by letting γ δ,φ ( n )( m ) = k iff there is some k (cid:48) ∈ N such that p σ n ( δ,φ ) p σ m ( δ,φ ) = p k (cid:48) and ( δ, φ, k (cid:48) , σ k ( δ, φ )) ∈ N . Note that n (cid:55)→ γ δ,ψ corresponds to the GNS representation of thenormed ∗ -algebra over Q ( i ) that corresponds to δ . Since the elements of N generate H ( δ, ψ ) andthe map ( δ, ψ ) (cid:55)→ ( γ δ,ψ ( n )( i ) | j ) δ,ψ URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 11 is Borel, it follows from Lemma 2.8 that there is a Borel functionΞ × C N → Γ( H ) : ( δ, ψ ) (cid:55)→ ˜ γ ( δ, ψ )such that ˜ γ ( δ, ψ ) ∈ Γ( H ) is conjugate to ( γ δ,φ ( n )) n ∈ N ∈ Γ( H ( δ, ψ )) for all δ, ψ .Since the map Ξ → K ( C N ) : δ (cid:55)→ S ( δ ) is Borel, by the Kuratowski–Ryll-Nardzewski theorem([13, Theorem 12.13]) there are Borel maps φ n : Ξ → C N such that for every δ ∈ Ξ the set φ n ( δ ),for n ∈ N , is dense in S ( δ ). Writing H = (cid:76) ∞ n =1 H n where H n are infinite dimensional Hilbertspaces, we may then by the above find a Borel map Ξ → Γ( H ) : δ → γ ( δ ) such that the restriction γ ( δ ) (cid:22) H n is conjugate to ˜ γ ( δ, φ n ). Since the sequence ( φ n ( δ )) is dense in S ( δ ), it follows that γ ( δ )is a faithful representation of the algebra corresponding to δ . (cid:3) Proof of Lemma 2.8. (1) is clear from the definition of Γ( H ). To see (2), first note that the Gram-Schmidt process provides orthonormal bases ( f xi ) i ∈ N for the H x such that f xi = i (cid:88) j =1 r xi,j e xj and the coefficient maps x (cid:55)→ r xi,j are Borel. Therefore the maps X (cid:55)→ C : ( g ( n )( x ) f xi | f xj )are Borel for all n, i, j ∈ N , and so we may in fact assume that ( e xi ) i ∈ N forms an orthonormal basisto begin with. But then if ( e i ) i ∈ N is an orthonormal basis a function ˆ g : X → Γ( H ) is defined byˆ g ( x )( n ) = S ⇐⇒ ( ∀ i, j )( Se i , e j ) = ( g ( n )( x ) e xi , e xj ) , and since this also provides a Borel description of the graph of ˆ g , ˆ g is Borel by [13, 14.12]. Finally,defining T x : H → H x to be the isometry mapping e i to e xi for each x provides the desiredconjugating map. (cid:3) Parameterizing unital C ∗ -algebras. We briefly discuss the parameterization of unital C ∗ -algebras. Define Γ u = { γ ∈ Γ : C ∗ ( γ ) is unital } . We will see (Lemma 3.14) that this set is Borel. We can similarly define ˆΓ u ⊆ ˆΓ, Ξ u ⊆ Ξ andˆΞ u ⊆ ˆΞ. However, as noted in 2.2, the setˆΓ A u ( H ) = { f : A u → B ( H ) : f is a unital ∗ -homomorphism } is Borel and naturally parameterizes the unital C ∗ -subalgebras of B ( H ). In analogy, we defineΞ A u = { f ∈ R A u : f defines a C ∗ -seminorm on A u with f (1) = 1 } , which is also Borel. Then a similar proof to that of Proposition 2.7 shows: Proposition 2.9.
The Borel sets ˆΓ A u and Ξ A u provide equivalent parameterizations of the unital C ∗ -algebras. In § A u and Ξ A u are also equivalent to Γ u (and therefore also ˆΓ u , Ξ u and ˆΞ u .)For future use, we fix once and for all an enumeration ( q n ) n ∈ N of all the formal Q ( i )- ∗ -polynomials(allowing constant terms), so that naturally A u = { q n : n ∈ N } . Also for future reference, we notethat Lemma 2.4 holds for Y = ˆΓ A u and Y = Ξ A u (the easy proof is left to the reader.) Basic maps and relations.
We close this section by making two simple, but useful, observa-tions pertaining to the parameterization Γ. While Borel structures of the weak operator topology,strong operator topology, σ -weak operator topology and σ -strong operator topology all coincide,the Borel structure of the norm topology is strictly finer. However, we have: Lemma 2.10.
Every norm open ball in B ( H ) is a Borel subset of Γ , and for every ε > the set { ( a, b ) : (cid:107) a − b (cid:107) < ε } is Borel. It follows that the maps B ( H ) → R : a (cid:55)→ (cid:107) a (cid:107) , B ( H ) → R : ( a, b ) (cid:55)→ (cid:107) a − b (cid:107) are Borel.Proof. Clearly { a : (cid:107) a (cid:107) > ε } is weakly open for all ε ≥
0. Hence norm open balls are F σ . (cid:3) Lemma 2.11.
The relations { ( γ, γ (cid:48) ) ∈ Γ × Γ : C ∗ ( γ ) ⊆ C ∗ ( γ (cid:48) ) } and { ( γ, γ (cid:48) ) ∈ Γ × Γ : C ∗ ( γ ) = C ∗ ( γ (cid:48) ) } are Borel.Proof. We have C ∗ ( γ ) ⊆ C ∗ ( γ (cid:48) ) ⇐⇒ ( ∀ n )( ∀ ε > ∃ m ) (cid:107) γ (cid:48) n − p m ( γ ) (cid:107) < ε, which is Borel by Lemma 2.10. (cid:3) Basic definability results
In this section we will show that a wide variety of standard C ∗ -algebra constructions correspondto Borel relations and functions in the spaces Γ and Ξ. Proposition 3.1. (1)
The relation (cid:45) on Γ , defined by γ (cid:45) δ if and only if C ∗ ( γ ) is isomorphic to a subalgebra of C ∗ ( δ ) , is analytic. (2) The relation (cid:39) Γ is analytic. In particular, Γ , ˆΓ , Ξ and ˆΞ are good standard Borel parame-terizations of the class of separable C ∗ -algebras. Before the proof of Proposition 3.1 we introduce some terminology and prove a lemma. Thefollowing terminology will be useful both here and later: We call Φ : N → N N a code for a ∗ -homomorphism C ∗ ( γ ) → C ∗ ( γ (cid:48) ) if for all m, n, k we have:(1) For each fixed m the sequence a m,k = p Φ( m )( k ) ( γ (cid:48) ), k ∈ N , is Cauchy. Write a m = lim k a m,k .(2) If p m ( γ ) + p n ( γ ) = p k ( γ ) then a m + a n = a k .(3) If p m ( γ ) p n ( γ ) = p k ( γ ) then a m a n = a k .(4) If p m ( γ ) ∗ = p k ( γ ) then a ∗ m = a k .(5) (cid:107) p m ( γ ) (cid:107) ≤ (cid:107) a m (cid:107) .We call Φ a code for a monomorphism if equality holds in (5). URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 13 Definitions of R hom , R mono and R iso . Let H and H be separable complex Hilbert spaces.Then it is easy to see that the relations R H ,H hom , R H ,H mono ⊆ Γ( H ) × Γ( H ) × ( N N ) N defined byR H ,H hom ( γ, γ (cid:48) , Φ) ⇐⇒ Φ is a code for a *-homomorphism C ∗ ( γ ) → C ∗ ( γ (cid:48) )R H ,H mono ( γ, γ (cid:48) , Φ) ⇐⇒ Φ is a code for a *-monomorphism C ∗ ( γ ) → C ∗ ( γ (cid:48) )are Borel. We let R H hom = R H,H hom and R H mono = R H,H mono for any Hilbert space H . If H , H or H areclear from the context or can be taken to be any (separable) Hilbert spaces then we will suppressthe superscript and write R hom and R mono . The following is immediate from the definitions: Lemma 3.2. If ( γ, γ (cid:48) , Φ) ∈ R hom then there is a unique homomorphism ˆΦ : C ∗ ( γ ) → C ∗ ( γ (cid:48) ) whichsatisfies ˆΦ( γ ( j )) = a j for all j ∈ N . If ( γ, γ (cid:48) , Φ) ∈ R mono then ˆΦ is a monomorphism.Proof. If R hom ( γ, γ (cid:48) , Φ) then p m ( γ ) (cid:55)→ a m is a *-homomorphism from a dense subalgebra of C ∗ ( γ ) into a subalgebra of C ∗ ( δ ). Since it is acontraction it extends to a *-homomorphism of ˆΦ : C ∗ ( γ ) → C ∗ ( γ (cid:48) ) onto a subalgebra of C ∗ ( γ ). IfR mono ( γ, γ (cid:48) , Φ) holds then ˆΦ is clearly a monomorphism. (cid:3)
We also define a relation R iso (we are suppressing H and H ) byR iso ( γ, γ (cid:48) , Φ) ⇐⇒ R mono ( γ, γ (cid:48) , Φ) ∧ ( ∀ m )( ∀ ε > ∃ k ∈ N )( ∀ n > k ) (cid:107) p Φ( m )( n ) ( γ ) − p m ( γ (cid:48) ) (cid:107) < ε. This relation states that Φ is a monomorphism and an epimorphism, and therefore an isomor-phism. It is Borel because R mono is Borel.
Proof of Proposition 3.1. (1) Clear, since γ (cid:45) γ (cid:48) ⇐⇒ ( ∃ Φ : N → N N ) R mono ( γ, γ (cid:48) , Φ) . (2) We have C ∗ ( γ ) (cid:39) C ∗ ( γ (cid:48) ) ⇐⇒ ( ∃ Φ : N → N N ) R iso ( γ, γ (cid:48) , Φ) , giving an analytic definition of (cid:39) Γ , and so Γ is a good parameterization. The last assertion followsfrom the equivalence of the four parameterizations. (cid:3) Remark . Note that the equivalence relation E on Γ defined by γ E δ if and only if there is aunitary u ∈ B ( H ) such that uC ∗ ( γ ) u ∗ = C ∗ ( δ ) is a proper subset of (cid:39) Γ and that E is inducedby a continuous action of the unitary group. We don’t know whether the relation (cid:39) Γ is an orbitequivalence relation induced be the action of a Polish group action on Γ, see discussion at the endof § Lemma 3.4.
The set Y of all γ ∈ Γ such that γ n , n ∈ N , is a Cauchy sequence (in norm) is Borel.The function Ψ : Y → B ( H ) that assigns the limit to a Cauchy sequence is Borel. Proof.
We have γ ∈ Y if and only if ( ∀ ε > ∃ m )( ∀ n ≥ m ) (cid:107) γ m − γ n (cid:107) < ε . By Lemma 2.10, theconclusion follows.It suffices to show that the graph G of Ψ is a Borel subset of B ( H ) N × B ( H ). But ( γ, a ) ∈ G if and only if for all ε > m such that for all n ≥ m we have (cid:107) γ m − a (cid:107) ≤ ε , which is byLemma 2.10 a Borel set. (cid:3) Directed systems, inductive limits, and R dir . A directed system of C*-algebras can becoded by a sequence ( γ i ) i ∈ N in Γ and a sequence Φ i : N → N N , for i ∈ N , such that( ∀ i ∈ N ) R hom ( γ i , γ i +1 , Φ i ) . The set R dir ⊆ Γ N × (( N N ) N ) N of codes for inductive systems is defined by(( γ i ) i ∈ N , (Φ i ) i ∈ N ) ∈ R dir ⇐⇒ ( ∀ i ∈ N ) R hom ( γ i , γ i +1 , Φ i )and is clearly Borel. Proposition 3.5.
There are Borel maps
LIM : R dir → Γ and Ψ i : R dir → ( N N ) N such that C ∗ (LIM(( γ i ) i ∈ N , (Φ i ) i ∈ N )) (cid:39) lim i →∞ ( C ∗ ( γ i ) , ˆΦ i ) and it holds that ( ∀ n ∈ N ) R hom ( γ n , LIM(( γ i ) i ∈ N , (Φ i ) i ∈ N ) , Ψ n (( γ i ) i ∈ N , (Φ i ) i ∈ N )) and ˆΨ n (( γ i ) i ∈ N , (Φ i ) i ∈ N ) : C ∗ ( γ ) → C ∗ (LIM(( γ i ) i ∈ N , (Φ i ) i ∈ N )) satisfies ˆΨ n +1 ◦ ˆΦ n = ˆΨ n , i.e. the diagram C ∗ ( γ n +1 ) ˆΨ n +1 (cid:47) (cid:47) LIM(( γ i ) i ∈ N , (Φ i ) i ∈ N ) C ∗ ( γ n ) ˆΦ n (cid:79) (cid:79) ˆΨ n (cid:52) (cid:52) commutes. We start by noting the simpler Lemma 3.6 below. The constant i sequence is denoted i . For( γ, γ (cid:48) , Φ) ∈ R hom define the function f : R hom → Γ by f ( γ, γ (cid:48) , Φ)( m ) = γ (cid:48) k if m = 3 k for k ≥ a if m = 2 k and lim i →∞ γ (cid:48) Φ( k )( i ) = a Lemma 3.6.
The function f introduced above is Borel and for all ( γ, γ (cid:48) , Φ) ∈ R hom we have C ∗ ( γ (cid:48) ) (cid:39) C ∗ ( f ( γ, γ (cid:48) , Φ)) . Moreover, for Ψ , Φ (cid:48) : N → N N defined by Φ (cid:48) ( m ) = 2 m and Ψ( m ) = 3 m for m ≥ , we have that R iso ( γ (cid:48) , f ( γ, γ (cid:48) , Φ) , Ψ) , R hom ( γ, f ( γ, γ (cid:48) , Φ) , Φ (cid:48) ) and ˆΨ ◦ ˆΦ = ˆΦ (cid:48) . URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 15 Proof of Proposition 3.5.
By Proposition 2.7 it will suffice to define LIM with the range in Ξ. Fix(( γ i ) i ∈ N , (Φ i ) i ∈ N ) ∈ R dir , let A = lim i →∞ ( C ∗ ( γ i ) , ˆΦ i ) , and let f i : C ∗ ( γ i ) → A be the connecting maps satisfying f i +1 ◦ ˆΦ i = f i . By Lemma 3.6 we mayassume that that for all m ∈ N the sequence Φ i ( m ), for i ∈ N , is constant. Let ϕ i ( m ) = Φ i ( m )(1),define ϕ i,j = ϕ i ◦ · · · ◦ ϕ j for j < i , and let β : N → N × N be a fixed bijection. Let ˜ γ ∈ Γ be definedby ˜ γ ( i ) = f β ( i ) ( γ β ( i ) ( β ( i ) )) . Then a code δ ∈ Ξ for ˜ γ is given by δ ( i ) = lim k →∞ (cid:107) p ϕ k,β ( i )0 ( β ( i ) ) ( γ k ) (cid:107) and if we define Ψ j (( γ i ) i ∈ N , (Φ i ) i ∈ N )( m )( n ) = k ⇐⇒ β ( k ) = j ∧ β ( k ) = m then Ψ j is a code for f j . (cid:3) Next we prove that most standard constructions and relations that occur in C ∗ -algebra theorycorrespond to Borel maps and relations in the parameterizations we have introduced. The firstlemma follows easily from the definitions, and we leave the proof to the reader. Lemma 3.7.
The following maps are Borel.(1) B ( H ) × B ( H ) → B ( H ) : ( a, b ) (cid:55)→ ab ,(2) B ( H ) × B ( H ) → B ( H ) : ( a, b ) (cid:55)→ a + b ,(3) B ( H ) × C → B ( H ) : ( a, λ ) (cid:55)→ λa ,(4) B ( H ) → B ( H ) : a (cid:55)→ a ∗ ,(5) B ( H ) × B ( H ) → B ( H ) ⊗ min B ( H ) : ( a, b ) (cid:55)→ a ⊗ b (where B ( H ) ⊗ min B ( H ) is identified with B ( H ) by fixing a ∗ -isomorphism),(6) B ( H ) × B ( H ) → M ( B ( H )) : ( a, b ) (cid:55)→ (cid:18) a b (cid:19) (where M ( B ( H )) is identified with B ( H ) byfixing a ∗ -isomorphism). Lemma 3.8.
The following subsets of B ( H ) , B ( H ) , and Γ are Borel.(1) { ( a, b ) : ab = ba } .(2) B ( H ) sa = { a : a = a ∗ } .(3) B ( H ) + = { a ∈ B ( H ) sa : a ≥ } .(4) P ( B ( H )) = { a ∈ B ( H ) : a is a projection } .(5) { a : a is a partial isometry } .(6) { a : a is invertible } .(7) { a : a is normal } .Proof. (4) Immediate since the maps a (cid:55)→ a − a and a (cid:55)→ a − a ∗ are Borel measurable.(5) Since a is a partial isometry if and only if a ∗ a and aa ∗ are both projections, this follows fromthe Borel-measurability of these maps and (4).(6) Let ξ n be a countable dense subset of the unit ball of B ( H ). Then a is invertible if and onlyif there is ε > (cid:107) aξ n (cid:107) ≥ ε and (cid:107) a ∗ ξ n (cid:107) ≥ ε for all n ([21, 3.2.6]).(7) Immediate since the map a (cid:55)→ [ a, a ∗ ] is Borel. (cid:3) Next we consider formation of the matrix algebra over a C ∗ -algebra. For this purpose, fixbijections β n : N → N n × n for each n . While the next Lemma is in some sense a special case of theLemma that follows it (which deals with tensor products) the formulation given below will be usedlater for the proof of Theorem 1.2. Lemma 3.9.
For each n ∈ N there are Borel functions M n : Γ( H ) → Γ( H n ) and θ n : Γ( H ) × ( N N ) n → N N such that(1) M n ( γ ) = ( γ β n ( l )(1 , · · · γ β n ( l )(1 ,n ) ... ... γ β n ( l )( n, · · · γ β n ( l )( n,n ) : l ∈ N ) (2) If ( γ, Ψ i ) ∈ R H hom for all i = 1 , . . . , n then ( γ, M n ( γ ) , θ n ( γ, Ψ , . . . , Ψ n )) ∈ R H,H n hom and θ n ( γ, Ψ , . . . , Ψ n )( k )( i ) = m = ⇒ p m ( M n ( γ )) = diag( γ (Ψ ( k )( i )) , . . . , γ (Ψ n ( k )( i ))) . That is, θ n ( γ, Ψ , . . . , Ψ n ) codes the diagonal embedding twisted by the homomorphisms ˆΨ i .Proof. (1) is clear. (2) follows by letting θ n ( γ, ψ , . . . , ψ n )( k )( i ) = m if and only if m is the leastsuch that p m ( M n ( γ )) = diag( γ (Ψ ( k )( i )) , . . . , γ ( ψ n ( k )( i ))) . (cid:3) Lemma 3.10.
There is a Borel-measurable map
Tensor : Γ × Γ → Γ such that C ∗ (Tensor( γ, δ )) ∼ = C ∗ ( γ ) ⊗ min C ∗ ( δ ) for all γ and δ in Γ .Moreover, there is a Borel-measurable map Tensor : Γ × Γ → Γ such that if ∈ C ∗ ( δ ) then C ∗ (Tensor ( γ, δ )) is the canonical copy of C ∗ ( γ ) inside C ∗ ( γ ) ⊗ min C ∗ ( δ ) .Proof. Fix a *-isomorphism Ψ : B ( H ) ⊗ min B ( H ) → B ( H ). DefineTensor( γ, δ ) m (2 n +1) = Ψ( γ m ⊗ δ n ) . Then Tensor is clearly Borel and the algebra generated by Tensor( γ, δ ) is C ∗ ( γ ) ⊗ min C ∗ ( δ ). Forthe moreover part, Tensor ( γ ) m = Ψ( γ m ⊗
1) clearly works. (cid:3)
It is not difficult to see that the set { γ ∈ Γ : 1 ∈ C ∗ ( γ ) } is Borel (cf. Lemma 3.14) but we shallnot need this fact. Lemma 3.11. If X is a locally compact Hausdorff space then there is a Borel measurable map Φ : Γ → Γ such that Φ( γ ) ∼ = C ( X, C ∗ ( γ )) for all γ ∈ Γ . In particular, letting X = (0 , , we conclude that there is a Borel map Φ such that Φ( γ ) is isomorphic to the suspension of C ∗ ( γ ) .Proof. This is immediate from Lemma 3.10 since C ( X, A ) ∼ = C ( X ) ⊗ min A . (cid:3) URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 17 Lemma 3.12.
There is a Borel function
Unit : Γ → Γ such that C ∗ (Unit( γ )) is isomorphic to theunitization of C ∗ ( γ ) .Proof. Fix a partial isometry v such that vv ∗ = 1 and v ∗ v is a projection onto a space of codimension1. Let Unit( γ ) = 1 and Unit( γ ) n +1 = v ∗ γ n v . Then C ∗ ( γ ) is as required. (cid:3) Effective enumerations.Lemma 3.13. (1)
There is a Borel map
Sa : Γ → Γ such that for every γ ∈ Γ the set { Sa( γ )( n ) : n ∈ N } is anorm-dense subset of the set of self-adjoint elements of C ∗ ( γ ) . (2) There is a Borel map
Un : Γ → Γ such that the set { Un( γ ) : n ∈ N } is norm-dense in the setof unitaries in C ∗ ( γ ) whenever C ∗ ( γ ) is unital. (3) There is a Borel map
Pos : Γ → Γ such that for every γ ∈ Γ the set { Pos( γ )( n ) : n ∈ N } is anorm-dense subset of the set of positive elements of C ∗ ( γ ) . (4) There is a Borel map
Proj : Γ → Γ such that for every γ ∈ Γ the set { Proj( γ )( n ) : n ∈ N } isa norm-dense subset of the set of projections of C ∗ ( γ ) .Proof. (1) Let Sa( γ )( n ) = ( p n ( γ ) + p n ( γ ) ∗ ) for all n . Clearly each Sa( γ )( n ) is self-adjoint. If a ∈ C ∗ ( γ ) is self-adjoint then (cid:107) a − Sa( γ )( n ) (cid:107) ≤ (cid:107) a − p n ( γ ) (cid:107) . Therefore the range of Sa is norm-dense subset of the set of self-adjoint elements of C ∗ ( γ ).(2) Let Un( γ )( n ) = exp( i Sa( γ )( n )).(3) Let Pos( γ )( n ) = p n ( γ ) ∗ p n ( γ ) for all n . Pick a positive a ∈ C ∗ ( γ ) and fix ε >
0. Pick b ∈ C ∗ ( γ ) such that a = b ∗ b . Let n be such that (cid:107) p n ( γ ) − b (cid:107) < ε/ (2 (cid:107) b (cid:107) ) and (cid:107) p n ( γ ) (cid:107) ≤ (cid:107) b (cid:107) . Then p n ( γ ) ∗ p n ( γ ) − a = ( p n ( γ ) ∗ − b ∗ ) p n ( γ ) + b ∗ ( p n ( γ ) − b )and the right hand side clearly has norm < ε .(4) Fix a function f : R → [0 ,
1] such that the iterates f n , n ∈ N , of f converge uniformly tothe function defined by g ( x ) = 0, x ≤ / g ( x ) = 1 for x ≥ / −∞ , / ∪ [3 / , ∞ ). Forexample, we can take f ( x ) = , x ≤ x , < x ≤ x − , < x ≤ − (1 − x ) / , < x ≤ , x > . The set X = B ( H ) sa is a Borel subset of B ( H ) by Lemma 3.8. Note that X ∩ { Pos( γ )( n ) : n ∈ N } is dense in X ∩ C ∗ ( γ ). Let Ψ : X → B ( H ) N be defined byΨ( a )( n ) = f n ( a ) . By Lemma 3.4 the set Y = { b ∈ X : Ψ( b ) is Cauchy } is Borel. For n such that Pos( γ )( n ) ∈ Y letProj( γ )( n ) be the limit of this sequence, and let Proj( γ )( n ) = 0 otherwise. By Lemma 3.4 again,Proj is Borel.Fix γ and n . Clearly, the operator Proj( γ )( n ) is positive and its spectrum is a subset of { , } .Therefore it is a projection in C ∗ ( γ ). We need to check that for every projection p ∈ C ∗ ( γ ) and ε > n such that (cid:107) Proj( γ )( n ) − p (cid:107) < ε . We may assume ε < /
4. Pick n so that (cid:107) Pos( γ )( n ) − p (cid:107) < ε . Since ε < /
4, the spectrum ofPos( γ )( n ) is included in ( − ε, ε ) ∪ (1 − ε, ε ) ⊆ ( − / , / ∪ (3 / , /
4) and therefore the sequence f j (Pos( γ )( n )), j ∈ N , converges to a projection, q . Clearly (cid:107) p − q (cid:107) < ε . (cid:3) Recall from 2.6 that Γ u denotes the set of γ ∈ Γ parameterizing unital C ∗ -algebras. From theprevious Lemma we now obtain: Lemma 3.14.
The set Γ u is Borel, and there is a Borel map u : Γ u → N such that Proj( γ )( u ( γ )) is the unit in C ∗ ( γ ) .Proof. For projections p and q we have that p ≤ q and p (cid:54) = q implies (cid:107) p − q (cid:107) = 1. Therefore C ∗ ( γ ) isunital if and only Proj( γ )( n ) is its unit for some n . Also, p is a unit in A if and only if pa = a = ap when a ranges over a dense subset of A . Therefore C ∗ ( γ ) is unital if and only if there is m suchthat for all n we have(3.1) Proj( γ )( m ) p n ( γ ) = p n ( γ ) Proj( γ )( m ) = p n ( γ ) . To define u : Γ u → N , simply let u ( γ ) = m if and only if m ∈ N is least such that 3.1 holds for all n ∈ N . (cid:3) Corollary 3.15.
The parameterization ˆΓ A u , Ξ A u , Γ u , ˆΓ u , Ξ u and ˆΞ u of unital separable C ∗ -algebrasall equivalent.Proof. It is clear from the previous Lemma and Propositions 2.6 and 2.7 that Γ u , ˆΓ u , Ξ u and ˆΞ u are equivalent standard Borel parameterizations. On the other hand, it is easy to see that Lemma2.4 hold for Y = ˆΓ A u , and so it is enough to show weak equivalence of Γ u and ˆΓ A u . In one direction,the natural map ˆΓ A u → Γ : f → γ ( f ) given by γ ( f )( n ) = f ( q n ) clearly works. The other directioncan be proven by a GNS argument analogous to the proof of proposition 2.7. (cid:3) Effros Borel structure. If X is a Polish space then F ( X ) denotes the space of all closedsubsets of X equipped with the σ -algebra generated by the sets { K ∈ F ( X ) : K ∩ U (cid:54) = ∅} for U ⊆ X open. This is a standard Borel space ([13, § F ( O )consisting of subalgebras of O (where O is the Cuntz algebra with two generators) is, by a resultof Kirchberg, a Borel space of all exact C*-algebras (see § A ⊆ X × Y and x ∈ X let A x denote the (vertical) section of A at x , that is, A x = { y :( x, y ) ∈ A } . Below (and later) we will need the following well-known fact (see [13, 28.8]) Lemma 3.16.
Let X and Y be Polish spaces and assume that A ⊆ X × Y is Borel and all sections A x are compact. Then the set A + = { ( x, A x ) : x ∈ X } is a Borel subset of X × F ( Y ) , and the map x (cid:55)→ A x is Borel. Coding states.
Roughly following [14, § C ∗ ( γ ). If φ is a functional on C ∗ ( γ ) then, being norm-continuous, it is uniquely determined by its restrictionto { p n ( γ ) : n ∈ N } . Also, writing ∆( r ) = { z ∈ C : | z | ≤ r } we have (cid:107) φ (cid:107) ≤ n we have φ ( p n ( γ )) ∈ ∆( (cid:107) p n ( γ ) (cid:107) ). Therefore we can identify φ with ˆ φ ∈ (cid:81) n ∆( (cid:107) p n ( γ ) (cid:107) ). Clearly,the set of ˆ φ such that φ is additive is compact in the product metric topology. Since φ is positive URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 19 if and only if φ ( p ∗ n ( γ ) p n ( γ )) ≥ n , the set of all states is also compact. Similarly, the set ofall traces is compact. By the obvious rescaling of the coordinates, we can identify (cid:81) n ∆( (cid:107) p n ( γ ) (cid:107) )with ∆ N (writing ∆ for ∆(1)). Consider the space K = K c (∆ N ) of compact subsets of ∆ N and itssubspace K conv of of compact convex subsets of ∆ N . Lemma 3.17.
With the above identifications, there are Borel maps S : Γ → K , P : Γ → K and T : Γ → K such that S ( γ ) is the set of all states on C ∗ ( γ ) , P ( γ ) is the closure of the set of all purestates on C ∗ ( γ ) and T ( γ ) is the set of all tracial states on C ∗ ( γ ) .Proof. For S and T this is obvious from the above discussion and Lemma 3.16. The existence of P can be proved by a proof similar to that of [14, Lemma 2.2]. (cid:3) Lemma 3.18.
There are Borel maps
State : Γ → (∆ N ) N , Pure : Γ → (∆ N ) N and Trace : Γ → (∆ N ) N such that State( γ )( m ) , for m ∈ N , is a dense subset of S ( γ ) , Pure( γ )( m ) , for m ∈ N is a densesubset of P ( γ ) and Trace( γ )( m ) , for m ∈ N , is a dense subset of T ( γ ) .Proof. For State and Trace this is a consequence of the previous lemma and the Kuratowski–Ryll-Nardzewski Theorem ([13, Theorem 12.13]). The construction of the map Pure, was given in [14,Corollary 2.3]. (cid:3) Choquet and Bauer simplexes
Let us first recall the pertinent definitions. All compact convex sets considered here will bemetrizable, and therefore without a loss of generality subsets of the Hilbert cube. For such S its extreme boundary , denoted ∂S , is the set of its extremal points. By the Krein–Milman theorem S is contained in the closure of the convex hull of ∂S . A metrizable Choquet simplex is a simplex S asabove with the following property: for every point x in S there exists a unique probability boundarymeasure µ (i.e., a measure concentrated on ∂S ) such that x is the barycentre of µ . This notionhas a number of equivalent definitions, see [1, § II.3]. The isomorphism relation in the category ofChoquet simplexes is affine homeomorphism.The extreme boundary of a Choquet simplex S is always G δ , and in the case that it is compact S is said to be a Bauer simplex . It is not difficult to see that in this case S is isomorphic to the space P ( ∂S ) of Borel probability measures on ∂S . In particular Bauer simplexes S and L are isomorphicif and only if their extreme boundaries ∂S and ∂L are homeomorphic.Let ∆ n denote the n -simplex ( n ∈ N ). Every metrizable Choquet simplex S can be representedas an inverse limit of finite-dimensional Choquet simplexes(4.1) S (cid:39) lim ← (∆ n i , ψ i ) , where and ψ i : ∆ n i → ∆ n i − is an affine surjection for each i ∈ N . This was proved in [18, Corollaryto Theorem 5.2] and we shall prove a Borel version of this result in Lemma 4.7.4.0.1. Order unit spaces.
Let (
A, A + ) be an ordered real Banach space. Here A + is a cone in A and the order is defined by a ≤ b if and only if b − a ∈ A + . Such a space is Archimedean if forevery a ∈ A the set { ra : r ∈ R + } has an upper bound precisely when a is negative , i.e., a ≤ A ∈ A is an order unit if for every a ∈ A there is r ∈ R + such that − r A ≤ a ≤ r A .We say that an Archimedean ordered vector space with a distinguished unit ( A, A + , A ) is an orderunit space , and define a norm on A by (cid:107) a (cid:107) = inf { r > − r A ≤ a ≤ r A } . Our interest in order unit spaces stems from the fact that the category of separable complete orderunit spaces is the dual category to the category of metrizable Choquet simplexes. For a Choquetsimplex S , the associated dual object is Aff( S ), the real-valued affine functions on S , with thenatural ordering and order unit set to be the constant function with value 1. Conversely, given anorder unit space ( A, A + , A ), the associated dual object is the space of positive real functionals φ on A of norm one, with respect to the weak*-topology. In the case of Bauer simplexes S there is alsoa natural identification of the complete separable order unit spaces Aff( S ) and C R ( ∂S ) obtainedby restriction. In particular, for the simplex ∆ n we haveAff(∆ n ) ∼ = ( R n +1 , ( R + ) n +1 , (1 , , . . . , . Setting e to be the origin in R n , the co-ordinate functions f k : ∆ n → R , 0 ≤ k ≤ n , given by theformula f k ( e i ) = (cid:26) i = k i (cid:54) = k on vertices and extended affinely, form a canonical basis for Aff(∆ n ).Let X and Y be separable order unit spaces with order units 1 X and 1 Y . Let as usual L ( X, Y )denote the set of linear, continuous maps, and let L ( X, Y ) = { T ∈ L ( X, Y ) : (cid:107) T (cid:107) ≤ } . The space L ( X, Y ) is a Polish space when given the strong topology. The set of order unit preserving mapsin L , L ou ( X, Y ) = { T ∈ L ( X, Y ) : ( ∀ x, x (cid:48) ∈ X ) x ≤ x (cid:48) = ⇒ T ( x ) ≤ T ( x (cid:48) ) ∧ T (1 X ) = 1 Y } is a closed subset of L ( X, Y ), and is therefore Polish in its own right. (Our definition of L ou involves some redundancy since it is a standard fact that T ∈ L ( X, Y ) such that T (1 X ) = 1 Y isautomatically order preserving.)4.1. Parameterizing metrizable Choquet simplexes and their duals.
The space Λ . If X = Aff( K ) and Y = Aff( L ) for metrizable Choquet simplexes K and L ,then L ou ( X, Y ) is the set of morphisms dual to the affine continuous maps from L to K . It followsfrom (4.1) that the separable complete order unit spaces all arise as direct limits of sequences R m φ −→ R m φ −→ R m φ −→ · · · with φ n ∈ L ou ( R m n , R m n +1 ). Since we can identify an operator in L ou ( R m n , R m n +1 ) with its matrix, L ou ( R m n , R m n +1 ) is affinely homeomorphic with a closed subspace of m n × m n +1 matrices. We cantherefore parameterize the separable complete order unit spaces (and therefore their duals) usingΛ = N N × (cid:89) ( m,n ) ∈ N L ou ( R m , R n )in the following way: each ( f, ψ ) ∈ Λ corresponds to the limit X ( f, ψ ) of the system R f (1) −→ ψ ( f (1) ,f (2)) R f (2) −→ ψ ( f (2) ,f (3)) R f (3) −→ ψ ( f (3) ,f (4)) · · · . Since Λ is a Polish space with respect to the product topology, we have what we will refer to asthe standard Borel space of metrizable Choquet simplexes . We note that our parameterization issimilar in spirit to that of Γ, as we identify our objects with something akin to a dense sequence.This is a good Borel parameterization (see Definition 2.1).
URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 21 The space Λ . The following Borel space of metrizable Choquet simplexes was essentiallydefined by Lazar and Lindenstrauss in [18] where the emphasis was put on Banach spaces X instead of the simplexes B ( X ∗ ). Another difference is that in [18] the authors studied a wider classof spaces whose dual is L .A simple analysis of an n × ( n + 1) matrix shows that, modulo permuting the basis of R n +1 ,every φ ∈ L ou ( R n , R n +1 ) is of the form(4.2) φ ( x , x , . . . , x n ) = ( x , x , . . . , x n , (cid:80) ni =1 a i x i )where 0 ≤ a i ≤ (cid:80) i a i = 1 is in L ou ( R n , R n +1 ).A representing matrix of a Choquet simplex is a matrix ( a ij ) ( i,j ) ∈ N in which all entries arenon-negative, (cid:80) ni =1 a in = 1, and a in = 0 for i > n . By the above, such a matrix codes a directedsystem R φ −→ R φ −→ R φ −→ · · · where φ n ( x , x , . . . , x n ) = ( x , x , . . . , x n , (cid:80) ni =1 a in x i ). A limit of this directed system is a Banachspace X and the unit ball of its dual is a Choquet simplex with respect to its weak*-topology. Thisis because an inverse limit of Choquet simplexes is again a Choquet simplex. We let Λ denote theset of all representing matrices, which is a closed set when viewed as a subset of [0 , ∞ ) N × N .On p. 184 of [18] the authors refer to the Borel space of representing matrices when they pointout that “It seems to be a very difficult problem to determine the set of all representing matrices ofa given separable infinite-dimensional predual of L ( µ ). We know the answer to this question onlyfor one such space, namely the space of Gurarii and even here the situation is not entirely clear.”Gurarii space is dual to the Poulsen simplex and the Lazar–Lindenstrauss characterization alludedto above implies that a dense G δ set of representing matrices corresponds to the Poulsen simplex.(By removing zeros, here we identify the matrix a in , i ≤ n ∈ N with an element of (cid:81) n [0 , n .) Thiscan be taken as a remark about the Borel complexity of certain set, close to the point of view ofthe present paper or of [14].4.1.3. The space Λ . Let δ n = 2 − n and for each n consider the set of all φ ∈ L ou ( R n , R n +1 ) of theform (4.2) such that all a i are of the form k − n for k ∈ N . Let F n be the set of all n × ( n +1) matricesrepresenting such φ . Modulo permuting basis of R n +1 , the set F n is δ n -dense in L ou ( R n , R n +1 ). Lemma 4.1.
For all m ≤ n in N and every Φ ∈ L ou ( R m , R n ) there are F i ∈ F i for m ≤ i ≤ n suchthat F n − ◦ · · · ◦ F m +1 ◦ F m is within − m from Φ composed with a permutation of the canonicalbasis of R n in the operator norm.Proof. The linear operator Φ is coded by an n × m -matrix ( a ij ) that has at least one entry equal to1 in each column. After possibly re-ordering the basis, we may then assume a ii = 1 for all i ≤ m .Furthermore, we can canonically write Φ as a composition of m − n operatorsΦ n − ◦ Φ n − ◦ · · · ◦ Φ m so that Φ k ∈ L ou ( R k , R k +1 ), and the last row of the matrix of Φ k is the k -th row of the matrix ofΦ padded with zeros. Now choose F n − , . . . , F m in (cid:81) n − k = m F k such that (cid:107) F k − Φ k (cid:107) < − k . Then F = F n − ◦ · · · ◦ F m is within 2 − m of Φ in the operator norm, as required. (cid:3) Let Λ be the compact metric space (cid:81) n F n . By identifying ψ ∈ Λ with (id , ψ ) ∈ Λ, one seesthat each element of Λ represents a Choquet simplex. We fix a well-ordering ≺ F of finite sequencesof elements of (cid:83) n F n , to be used in the proof of Lemma 4.7. The space K Choq . Recall that K conv is the space of all compact convex subsets of the Hilbertcube. Let K Choq denote the space of all Choquet simplexes in K ∈ K conv . In Lemma 4.7 we shallshow that K Choq is a Borel subspace of K , and therefore K Choq is the ‘natural’ parameterization ofChoquet simplexes.4.1.5.
Our Borel parameterizations of Choquet simplexes are weakly equivalent.
Weak equivalenceof Borel parameterizations was defined in (4’) of Definition 2.1.
Proposition 4.2.
The four Borel parameterizations of Choquet simplexes introduced above, Λ , Λ , Λ , and K Choq , are all weakly equivalent.
A proof of Proposition 4.2 will take up the rest of this section. Clearly the space K conv is a closedsubset of K c (∆ N ). In the following consider the Effros Borel space F (C R (∆)) of all closed subsetsof C R (∆) (see § peaked partition of unity in an order-unit space ( A.A + , A ) is a finite set f , . . . , f n of positive elements of A such that (cid:80) i f i = 1 A and (cid:107) f i (cid:107) = 1 for all i . A peaked partition of unity P (cid:48) refines a peaked partition of unity P if every element of P is a convex combination of the elementsof P (cid:48) .We shall need two facts about real Banach spaces. For a separable Banach space X let S ( X )denote the space of closed subspaces of X , with respect to the Effros Borel structure ( § S (C R (∆)) is universal for separable Banach spaces, and therefore this spacewith respect to its Effros Borel structure can be considered as the standard Borel space of separableBanach spaces (see Lemma 4.4). Consider the space K conv ⊆ K c (∆ N ) of compact convex subsetsof the Hilbert cube, ∆ N . With respect to the Borel structure induced by the Hausdorff metric, thisis the standard Borel space of all compact convex metrizable spaces. For a Banach space X let B ( X ∗ ) denote the unit ball of the dual of X , with respect to the weak*-topology. Then B ( X ∗ ) isa compact convex space, and it is metrizable if X is separable. The idea in the following is takenfrom of the proof of [14, Lemma 2.2]. Lemma 4.3. If X is a separable Banach space then there is a Borel map Φ : S ( X ) → K conv suchthat Φ( Y ) is affinely homeomorphic to the unit ball B ( Y ∗ ) of Y ∗ , with respect to its weak*-topology.Proof. By Kuratowski–Ryll-Nardzewski’s theorem ([13, Theorem 12.13]) there are Borel f n : S ( X ) → X for n ∈ N such that { f n ( Y ) : n ∈ N } is a dense subset of Y for every Y .Fix an enumeration F n = ( r n,i : i ≤ k n ), for n ∈ N , of finite sequences of rationals. Define h n : S ( X ) → X by h n ( Y ) = k n (cid:88) i =1 r n,i f i ( Y ) . Then { h n ( Y ) : n ∈ N } is a dense linear subspace of Y for each Y ∈ S ( X ).Let ∆( Y ) = (cid:81) n [ −(cid:107) h n ( Y ) (cid:107) , (cid:107) h n ( Y ) (cid:107) ]. Let K ( Y ) be the set of all φ ∈ ∆( Y ) such that(*) F i + F j = F l (where the sum is taken pointwise) implies φ ( i ) + φ ( j ) = φ ( l ), for all i, j and l .Such a φ defines a functional of norm ≤ Y , and therefore extends to anelement of B ( Y ∗ ). Moreover, every functional in B ( Y ∗ ) is obtained in this way. Therefore the setof φ satisfying (*) is affinely homeomorphic to B ( Y ∗ ).It remains to rescale K ( Y ). Let Φ( Y ) = { φ ∈ ∆ N : ( φ ( n ) (cid:107) h n ( Y ) (cid:107) ) n ∈ N ∈ K ( Y ) } . Then φ ∈ Φ( Y )if and only if φ ( i ) (cid:107) h i ( Y ) (cid:107) + φ ( j ) (cid:107) h j ( Y ) (cid:107) = φ ( l ) (cid:107) h l ( Y ) (cid:107) URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 23 for all triples i, j, l satisfying F i + F j = F l (a condition not depending on Y ).Since the map y (cid:55)→ (cid:107) y (cid:107) is continuous, the map Y (cid:55)→ Φ( Y ) is Borel, and clearly Φ( Y ) is affinelyhomeomorphic to K ( Y ) and therefore to B ( Y ∗ ). (cid:3) Lemma 4.4.
There is a Borel map
Ψ : K conv → S (C R (∆)) such that the Banach spaces Aff( K ) and Ψ( K ) are isometrically isomorphic for all K .Proof. Identify ∆ N with (cid:81) n [ − /n, /n ]. Consider the compatible (cid:96) metric d on ∆ N and the set Z = { ( K, x, y ) : K ∈ K conv , x ∈ ∆ N , y ∈ K, and d ( x, y ) = inf z ∈ K d ( x, z ) } . Since the map (
K, x ) (cid:55)→ inf z ∈ K d ( x, z ) is continuous on { K ∈ K conv : K (cid:54) = ∅} , this set is closed.Also, for every pair K, x there is the unique point y such that ( K, x, y ) ∈ Z (e.g., [21, Lemma 3.1.6]).By compactness, the function χ that sends ( K, x ) to the unique y such that ( K, x, y ) ∈ Z iscontinuous. Fix a continuous surjective map η : ∆ → ∆ N . Then χ K ( x ) = χ ( K, η ( x )) defines acontinuous surjection from ∆ onto K and K (cid:55)→ χ K is a continuous map from K conv into C (∆ , ∆ N )with respect to the uniform metric. The set Y = { ( K, f ) ∈ K conv × C R (∆) : for some g ∈ Aff( K ) } is closed. To see this, note that ( K, f ) / ∈ Y iff one of the following two conditions happens:(1) There are x and y such that f ( x ) (cid:54) = f ( y ) but χ ( K, f )( x ) = χ ( K, f )( y ), or(2) There are x, y, z and 0 < t < f ( tx + (1 − t ) y ) (cid:54) = z but tχ ( K, f )( x ) + (1 − t ) χ ( K, f )( y ) = χ ( K, f )( z ) . We need to prove that the map that sends K to Y K = { f : ( K, f ) ∈ Y} is Borel. Since Y K isclearly isometric to Aff( K ), this will conclude the proof.Let g n , for n ∈ N , be a countable dense subset of C R (∆). By compactness h n ( K ) = ( g n (cid:22) K ) ◦ χ K ◦ η is a continuous map from K conv to C R (∆) such that h n ( K ) ∈ Y K . Moreover, the set { h n ( K ) : n ∈ N } is dense in Y K for every K . Since Y K (cid:54) = ∅ , we conclude that the map Ψ( K ) = Y K is Borel. Thisfollows by [13, 12.14] or directly by noticing that Ψ − ( { X ∈ S (C R (∆)) : X ∩ U (cid:54) = ∅} = (cid:84) n h − ( U )is Borel for every open U ⊆ C R (∆). (cid:3) Let Ψ : K conv → S (C R (∆)) be the Borel-measurable map that sends K to Aff( K ) ⊆ C R (∆) fromLemma 4.4. For every K and n the setPPU n ( K ) ⊆ (C R (∆)) n of all n -tuples in Ψ( K ) forming a peaked partition of unity is closed, by compactness of K .The following lemma is a reformulation of (4.1). Lemma 4.5.
For a metrizable compact convex set K the following are equivalent.(1) K is a Choquet simplex,(2) for every finite F ⊆ Aff( K ) , every (cid:15) > and every peaked partition of unity P in Aff( K ) there is a peaked partition of unity P (cid:48) that refines P and is such that every element of F iswithin ε of the span of P (cid:48) . (cid:3) Another equivalent condition, in which (2) is weakened to approximate refinement, followsfrom [31] and it will be reproved during the course of the proof of Lemma 4.7 below.
Lemma 4.6.
The map from K conv to F (C R (∆) n ) that sends K to PPU n ( K ) is Borel for everyfixed n .Proof. The set of all (
K, f , f , . . . , f n ) ∈ K conv × (C R (∆)) n such that f i ∈ Ψ( K ) for 1 ≤ i ≤ n and (cid:80) i ≤ n f i ≡ K, f , . . . , f n ) such that f i ∈ Ψ( K ) for all i ≤ n , and the conclusion follows. (cid:3) By [13, Theorem 12.13] or the proof of Lemma 4.4 and the above we have Borel maps h n : K conv → C R (∆) such that { h n ( K ) : n ∈ N } is a dense subset of Ψ( K ), and Borel maps P i,n : K conv → (C R (∆)) n , for i ∈ N , such that { P i,n ( K ) : i ∈ N } is a dense subset of PPU n ( K ), for every K ∈ K conv . Also fix h i : K → ∆ N such that { h i ( K ) : i ∈ N } is a dense subset of K for all K . Lemma 4.7.
The set K Choq is a Borel subset of K . Moreover, there is a Borel map Υ : K Choq → Λ such that Υ( K ) is a parameter for K .Proof. We shall prove both assertions simultaneously. Let ε i = i − − i − .Fix K ∈ K Choq for a moment. Let us say that a partition of unity
P ε -refines a partition ofunity P (cid:48) if every element of P (cid:48) is within ε of the span of P . By Lemma 4.5, there are sequences d ( j ) = d ( j, K ), i ( j ) = i ( j, K ) and n ( j ) = n ( j, K ), for j ∈ N , such that for each j we have(1) P i ( j +1) ,n ( j +1) ( K ) is in PPU d ( j ) ( K ),(2) { h i ( K ) (cid:22) K : i ≤ j } and the restriction of all elements of P i ( j ) ,n ( j ) ( K ) to K are within ε j ofthe rational linear span of the restrictions of elements of P i ( j +1) ,n ( j +1) ( K ) to K ,(3) i ( j + 1) , n ( j + 1) is the lexicographically minimal pair for which (1) and (2) hold.The set of all triples ( K, ( i ( j ) : j ∈ N ) , ( n ( j ) : j ∈ N )) such that (1) and (2) hold is Borel. Since afunction is Borel if and only if its graph is Borel ([13]), the function sending K to (( i ( j, K ) , n ( j, K )) : j ∈ N ) is Borel.Still having K fixed, let us write P j for P i ( j,K ) ,n ( j,K ) ( K ). Since each f ∈ P j is within ε j of thespan of P j +1 , [31, Lemma 2.7] implies there is an isometry Φ j : span( P j ) → span( P j +1 ) such that (cid:107) Φ j ( f ) − f (cid:107) < − j for all f ∈ span( P j ). Using Lemma 4.1, we can fix the ≺ F -least composition ofoperators in (cid:83) n F n (see § ψ ( j ), that 2 − j -approximates Φ j in the operator norm. This definesan element ψ of Λ . Again, the function that associates ψ to K is Borel since its graph is a Borelset.It remains to prove that the direct limit of R d ( j ) , for j ∈ N , determined by ψ is isometric toAff( K ). For every fixed k the sequence of linear operators ψ ( k + j ) ◦ ψ ( k + j − ◦ . . . ψ ( k ) for j ∈ N forms a Cauchy sequence in the supremum norm. Therefore the image of P k under this sequenceconverges to a peaked partition of unity, denoted by Q k , of Aff( K ). Then Q k , for k ∈ N , form arefining sequence of peaked partitions of unity of Aff( K ) such that the span of (cid:83) k Q k is dense inAff( K ). Thus with the dependence of ψ on K understood, we have that Υ( K ) := ψ is the requiredparameter for K in Λ . (cid:3) The following Lemma will only be used later in Section 5, but we include it here as it fitsthematically in this section.
Lemma 4.8.
There is a Borel map
Ψ : K c (∆ N ) → Λ such that Ψ( K ) represents a Choquet simplexaffinely homeomorphic to the Bauer simplex P ( K ) .Proof. By Lemma 4.7 it suffices to define a Borel map Ψ : K c (∆ N ) → K Choq so that Ψ ( K )is affinely homeomorphic to P ( K ) for all K . For each K ∈ K c (∆ N ) the set P ( K ) is affinely URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 25 homeomorphic to a closed convex subset Y K of P (∆ N ), by identifying each measure ν on K withits canonical extension ν (cid:48) to ∆ N , ν (cid:48) ( A ) = ν ( A ∩ K ). Moreover, the map K (cid:55)→ Y K is continuous withrespect to the Hausdorff metric. Fix an affine homeomorphism of P (∆ N ) into ∆ N . For example, if f n , for n ∈ N , is a sequence uniformly dense in { f : ∆ N → ∆ } then take ν (cid:55)→ ( (cid:82) f n dν : n ∈ N ). Bycomposing the map K (cid:55)→ Y K with this map we conclude the proof. (cid:3) Proof of Proposition 4.2.
A Borel homomorphism from the parametrization K Choq to Λ was givenin Lemma 4.7. If ψ ∈ Λ then (possibly after permuting the basis of R n +1 ) each ψ ( n ) defines a n , . . . , a nn as in § into Λ . This map is continuous, and even Lipschitz in the sense that ψ ( n ) determines all a in for i ≤ n . Similarly, every representing matrix in Λ canonically defines a directed system in Λ.We therefore only need to check that there is a Borel homomorphism from Λ to K Choq .Given ( f, ψ ) ∈ Λ, we define K = K ( f, ψ ) as follows. With f (0) = 0 let k n = (cid:80) ni =0 f ( i ). For a ∈ R N and n ≥ a n = a (cid:22) [ k n , k n +1 ) and identify a ∈ R N with ( a n ) n ∈ N . Let B = { a ∈ R N : ( ∀ n ) ψ n ( a n ) = a n +1 } . Then B = B ( f, ψ ) is a separable subspace of (cid:96) ∞ closed in the producttopology. Also, ( f, ψ ) (cid:55)→ B ( f, ψ ) ∩ ∆ N is a continuous map from Λ into the hyperspace of ∆ N , andtherefore the map ( f, ψ ) (cid:55)→ B ( f, ψ ) is a Borel map from Λ into F ( R N ).Let K ( f, ψ ) denote the unit ball B ∗ ( f, ψ ) of the dual of the Banach space B ( f, ψ ). Whenequipped with the weak*-topology, K ( f, ψ ) is affinely homeomorphic to the Choquet simplex rep-resented by ( f, ψ ). We complete the proof by applying Lemma 4.3. (cid:3) The isomorphism relation for AI algebras
Recall that an approximately interval (or AI ) C ∗ -algebra is a direct limit A = lim −→ ( A i , φ i ) , where, for each i ∈ N , A i ∼ = F i ⊗ C([0 , ∗ -algebra F i and φ i : A i → A i +1 is a ∗ -homomorphism. In this section we will prove the following (Λ is the space defined in § X ( f, ψ ) was introduced in § Theorem 5.1.
There is a Borel function ζ : Λ → Γ such that for all ( f, ψ ) ∈ Λ ,(1) C ∗ ( ζ ( f, ψ )) is a unital simple AI algebra.(2) ( K ( C ∗ ( ζ ( f, ψ )) , K +0 ( C ∗ ( ζ ( f, ψ )) , (cid:39) ( Q , Q + , and K ( C ∗ ( ζ ( f, ψ ))) (cid:39) { } .(3) If T is the tracial state simplex of C ∗ ( ζ ( f, ψ )) then Aff( T ) (cid:39) X ( f, ψ ) . We note that Theorem 5.1 immediately implies Theorem 1.2:
Corollary 5.2.
The following relations are Borel reducible to isomorphism of simple unital AI algebras:(1) Affine homeomorphism of Choquet simplexes.(2) Homeomorphisms of compact Polish spaces.(3) For any countable language L , the isomorphism relation (cid:39) Mod( L ) on countable models of L .Moreover, isomorphism of simple unital AI algebras is not classifiable by countable structures, andis not a Borel equivalence relation.Proof. For (1), let ζ be as in Theorem 5.1. Since simple unital AI algebras are classified by theirElliott invariant and since ( Q , Q + ,
1) has a unique state, it follows that ( f, ψ ) (cid:39) Λ ( f (cid:48) , ψ (cid:48) ) if andonly if C ∗ ( ζ ( f, ψ )) (cid:39) C ∗ ( ζ ( f (cid:48) , ψ (cid:48) )). For (2), note that by Lemma 4.8, homeomorphism of compact subsets of [0 , N is Borel reducibleto affine homeomorphism in Λ.(3) follows from (2) and [9, 4.21], where it was shown (cid:39) L is Borel reducible to homeomorphism.It was shown in [9, 4.22] that homeomorphism of compact subsets of K is not classifiable bycountable structures, and so by (2) neither is isomorphism of AI algebras. Finally, it was shownin [7] that (cid:39) Mod( L ) is not Borel when L consists of just a single binary relation symbol, and so itfollows from (3) that isomorphism of simple unital AI algebras is not Borel. (cid:3) The strategy underlying the proof of Theorem 5.1 is parallel to the main argument in [28]. As afirst step, we prove the following:
Lemma 5.3.
There is a Borel map ς : Λ → L ou ( C R [0 , N such that for all ( f, ψ ) ∈ Λ we have (5.1) X ( f, ψ ) (cid:39) lim( C R [0 , , ς ( f, ψ ) n ) . Proof.
Let f , ∈ C R [0 ,
1] be the constant 1 function, and for each n > ≤ i ≤ n −
1, let f n,i : [0 , → R be the function such that f n,i (cid:18) jn − (cid:19) = (cid:26) j = i j (cid:54) = i and which is piecewise linear elsewhere. Then P n = { f n,i : 0 ≤ i ≤ n } is a peaked partition ofunity. For each n , let η n : R n → C R [0 ,
1] be the linear map given on the standard basis ( e i ) of R n by η n ( e i ) = f n,i , and let β n : C R [0 , → R n be given by β n ( f ) i = f ( in − ). Then η n and β n are orderunit space homomorphisms and β n ◦ η n = id R n . Define ς ( f, ψ ) n = η f ( n +1) ◦ ψ ( f ( n ) , f ( n + 1)) ◦ β f ( n ) and note that ς is continuous, and so it is Borel. Since the diagram C R ([0 , ς ( f,ψ ) (cid:47) (cid:47) β f (1) (cid:15) (cid:15) C R [0 , ς ( f,ψ ) (cid:47) (cid:47) β f (2) (cid:15) (cid:15) C R [0 , ς ( f,ψ ) (cid:47) (cid:47) β f (3) (cid:15) (cid:15) · · · R f (1) ψ ( f (1) ,f (2)) (cid:47) (cid:47) R f (2) ψ ( f (2) ,f (3)) (cid:47) (cid:47) R f (3) ψ ( f (3) ,f (4)) (cid:47) (cid:47) · · · commutes, (5.1) holds. (cid:3) Before proceeding, we fix our notation and collect the key results from [28] that we need. Weidentify C [0 , ⊗ M n ( C ) and M n ( C [0 , φ : M n ( C [0 , → M m ( C [0 , standard homomorphism when there are continuous functions f , . . . , f mn : [0 , → [0 , φ ( g ) = diag( g ◦ f , . . . , g ◦ f mn ). Following [28], we will call the sequence f , . . . , f mn the characteristic functions of the standard homomorphism φ . The tracial state space of M n ( C [0 , ,
1] (see [28, p. 606]), and so wecanonically identify Aff( T ( M n ( C [0 , C R [0 , Lemma 5.4 (Thomsen) . (1) Any AI algebra can be represented as an inductive limit lim n ( M n ( C [0 , , φ n ) , where each φ n is a standard homomorphism. URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 27 (2) If φ : M n ( C [0 , → M m ( C [0 , is a standard homomorphism with characteristic functions f , . . . , f mn , then the induced order unit space homomorphism ˆ φ : C R [0 , → C R [0 , (underthe natural identification with the tracial state spaces) is given by ˆ φ ( g ) = nm mn (cid:88) i =1 g ◦ f i . (3) Let φ i , ψ i ∈ L ou ( C R [0 , be order unit morphisms ( i ∈ N ) and let δ i ∈ R + be a sequencesuch that (cid:80) ∞ i =1 δ n < ∞ . Suppose there are finite sets F k ⊆ C R [0 , such that(a) F k ⊆ F k +1 for all k ∈ N ;(b) (cid:83) k F k has dense span in C R [0 , ;(c) for all f ∈ F k there are g, h ∈ F k +1 such that (cid:107) φ i ( f ) − g (cid:107) , (cid:107) ψ i ( f ) − h (cid:107) ≤ δ k +1 for all i ≤ k ;(d) for all f ∈ F k we have (cid:107) φ k ( f ) − ψ k ( f ) (cid:107) ≤ δ k .Then lim → ( C R [0 , , φ i ) and lim → ( C R [0 , , ψ i ) are isomorphic as order unit spaces.(4) For any order unit homomorphism ψ : C R [0 , → C R [0 , , f ∈ C R [0 , , finite F ⊆ C R [0 , , n, k ∈ N and ε > there is m = m nk ∈ N and continuous f , . . . f mn : [0 , → [0 , suchthat for all g ∈ F we have (5.2) (cid:107) ψ ( g ) − nm mn (cid:88) i =1 g ◦ f i (cid:107) ∞ < ε. Proof. (1) and (2) are simply restatements of Lemma 1.1 and Lemma 3.5 in [28], while (3) followsimmediately from [28, Lemma 3.4]. For (4), note that by the Krein-Milman type theorem [28,Theorem 2.1], we can find a multiple d of k and continuous ˜ f , . . . , ˜ f N : [0 , → [0 ,
1] such that (cid:107) ψ ( g ) − (cid:80) Ni =1 n i d ( g ◦ ˜ f i ) (cid:107) ∞ < ε where 1 ≤ n i ≤ d satisfy (cid:80) Ni =1 n i = d . Let m = dn and let f , . . . , f mn be the list of functions obtained by repeating n times ˜ f , then n times ˜ f , etc. Then (5.2) isclearly satisfied. (cid:3) For the next lemma we refer back to § H ,H hom and the functions M n : Γ( H ) → Γ( H n ) and θ n : Γ( H ) × ( N N ) n → N N . Lemma 5.5.
View C [0 , as multiplication operators on H = L ([0 , . Then there is an element γ ∈ Γ( H ) such that C ∗ ( γ ) is equal to C [0 , and such that there are Borel maps d N : L ou ( C R [0 , N → N and Φ N : L ou ( C R [0 , N → N N for all N ∈ N , so that for all (cid:126)ς ∈ L ou ( C R [0 , N we have: ( I ) For all N ∈ N we have ( M d N ( (cid:126)ς ) ( γ ) , M d N +1 ( (cid:126)ς ) ( γ ) , Φ N ( (cid:126)ς )) ∈ R H dN ( (cid:126)ς ) ,H dN +1( (cid:126)ς ) hom . ( II ) The limit A (cid:126)ς = lim N ( C ∗ ( M d N ( (cid:126)ς ) i ( γ )) , ˆΦ N ( (cid:126)ς )) is a unital simple AI algebra, which satisfies ( K ( A (cid:126)ς ) , K +0 ( A (cid:126)ς ) , [1 A (cid:126)ς ]) (cid:39) ( Q , Q + , , K ( A (cid:126)ς ) = { } and Aff( T ( A (cid:126)ς )) (cid:39) lim N ( C R [0 , , (cid:126)ς N ) . Proof.
Fix a sequence of continuous functions λ n : [0 , → [0 ,
1] which is dense in C ([0 , , [0 , λ ( x ) = x and λ n enumerates all rational valued constant functions with infiniterepetition. Also fix a dense sequence g n ∈ C R [0 , n ∈ N , closed under composition with the λ n (i.e., for all i, j ∈ N there is k ∈ N such that g i ◦ λ j = g k .)Pick γ ∈ Γ( H ) to consist of the operators on H that correspond to multiplication by the g n . Each λ n induces an endomorphism ψ n,m of C ∗ ( M m ( γ )) by entry-wise composition. Let Ψ n,m : N → N N enumerate a sequence of codes corresponding to the ψ n,m . These may even be chosen so thatΨ n,m ( l ) is always a constant sequence since we assumed that the sequence ( g n ) is closed undercomposition with the λ k .Define for each N ∈ N a relation R N ⊆ L ou ( C R [0 , × N × N × Q + × N × N < N by R N ( ψ, n, k, ε, m, t ) ⇐⇒ mnk ∈ N ∧ length( t ) = mn ∧ t (1) = 1 ∧ t (2) = 2 N ∧ ( ∀ j ≤ N ) (cid:107) ψ ( g j ) − nm mn (cid:88) i =1 g j ◦ λ t ( i ) (cid:107) ∞ < ε Note that this is an open relation in the product space when L ou ( C R [0 , N , Q + and N < N have the discrete topology.) By Lemma 5.4.(4) it holds that for all ψ , n , k and ε there is m and t such that R N ( ψ, n, k, ε, m, t ) holds. (Note that this still holds although wehave fixed the first two elements of the sequence t , since m can be picked arbitrarily large.) Let t N ( ψ, n, k, ε ) be the lexicographically least t ∈ N < N such that R N ( ψ, n, k, ε, n length( t ) , t ) holds.We let m N ( ψ, n, k, ε ) = n length( t ), and note that t N and m N define Borel functions.Fix a sequence ( δ i ) i ∈ N in Q + such that (cid:80) ∞ i =1 δ i < ∞ . Let q i ∈ N enumerate the primes witheach prime repeated infinitely often. We can then define Borel functions G N : L ou ( C R [0 , N → N , d N : L ou ( C R [0 , N → N , d N : L ou ( C R [0 , N → N and s N : L ou ( C R [0 , → N < N recursively suchthat the following is satisfied:( A ) G , d and d are the constant 1 functions, s is constantly the empty sequence.( B ) G N +1 ( (cid:126)ς ) is the least natural number k such that for all i ≤ N and j ≤ G N ( (cid:126)ς ) there are j , j ≤ k such that (cid:107) d N (cid:88) l =1 g j ◦ λ s i ( (cid:126)ς ) l − g j (cid:107) ≤ δ N and (cid:107) (cid:126)ς i ( g j ) ◦ λ s i ( (cid:126)ς ) l − g j (cid:107) ≤ δ N . ( C ) d N +1 ( (cid:126)ς ) = m G N +1 ( (cid:126)ς ) ( (cid:126)ς N +1 , d N ( (cid:126)ς ) , q · · · , q N , δ N +1 ).( D ) s N +1 ( (cid:126)ς ) = t G N +1 ( (cid:126)ς ) ( (cid:126)ς N +1 , d N ( (cid:126)ς ) , q · · · q N , δ N +1 ).( E ) d N +1 ( (cid:126)ς ) = d N +1 ( (cid:126)ς ) d N ( (cid:126)ς ) = length( s N +1 ( (cid:126)ς )).Note that d N takes integer values by the definition of d N . DefineΦ N ( (cid:126)ς ) = θ d N +1 ( (cid:126)ς ) ( M d N ( (cid:126)ς ) ( γ ) , Ψ s N +1 ( (cid:126)ς ) ,d N ( (cid:126)ς ) , . . . , Ψ s N +1 ( (cid:126)ς ) d N +1( (cid:126)ς ) ,d N ( (cid:126)ς ) ) . Then Φ N and d N are Borel functions for all N ∈ N , and (I) of the Lemma holds by definition of θ n .We proceed to prove that (II) also holds. Fix (cid:126)ς ∈ L ou ( C R [0 , N . Note that the inductive system( C ∗ ( M d N ( (cid:126)ς ) ( γ ) , ˆΦ N ( (cid:126)ς )) is isomorphic to the system ( M d N ( (cid:126)ς ) ( C [0 , , φ N ) where φ N ( f ) = diag( f ◦ λ s N +1 ( (cid:126)ς ) , . . . , f ◦ λ s N +1 ( (cid:126)ς ) d N +1( ς ) ) . URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 29 Since each natural number divides some d N ( (cid:126)ς ) we have( K ( A (cid:126)ς ) , K +0 ( A (cid:126)ς ) , [1 A (cid:126)ς ]) (cid:39) ( Q , Q + , K ( A (cid:126)ς ) = { } since [0 ,
1] is contractible.To establish that Aff( T ( A (cid:126)ς )) (cid:39) lim i ( C R [0 , , (cid:126)ς i ) we apply Lemma 5.4. By Lemma 5.4.(2) theorder unit space morphism induced by φ N is given byˆ φ N ( f ) = 1 d N +1 ( (cid:126)ς ) d N +1 ( (cid:126)ς ) (cid:88) i =1 f ◦ λ s N +1 ( (cid:126)ς ) i . Letting F N = { g i : i ≤ G N ( (cid:126)ς ) } , it is clear that (a) and (b) of Lemma 5.4.(3) are satisfied. That (c)of 5.4.(3) then also is satisfied for the sequences ˆ φ N , (cid:126)ς N ∈ C R [0 ,
1] follows from property (B) above.Finally, 5.4.(3).(d) holds by (D) and the definition of t N and R N . Thuslim i ( C R [0 , , (cid:126)ς i ) (cid:39) lim i ( C R [0 , , ˆ φ i ) (cid:39) Aff( T ( A (cid:126)ς )) . It remains only to verify that A (cid:126)ς is simple. For this we need only prove that if 0 (cid:54) = f ∈ M d N ( (cid:126)ς ) ( C [0 , t ∈ [0 ,
1] we have φ N,j ( f ) := ( φ j − ◦ φ j − ◦ · · · ◦ φ N ) ( f )is nonzero at t for some (and hence all larger) j ≥ N . By the definition of the sequence ( λ n ), thereis some j ≥ l such that f ◦ λ j (cid:54) = 0. By the definition of the relations R n , f is a direct summandof φ l,j ( f ), and so the constant function f ◦ λ j (cid:54) = 0 is a direct summand of φ N,j +1 . This implies φ N,j +1 ( f )( t ) (cid:54) = 0 for each t ∈ [0 , (cid:3) Proof of Theorem 5.1.
Combine Lemma 5.3 with Lemma 5.5. (cid:3)
Corollary 5.6.
There is a Borel measurable map Φ from { γ : C ∗ ( γ ) is unital and abelian } into { γ : C ∗ ( γ ) is simple and unital AI } such that C ∗ ( γ ) ∼ = C ∗ ( γ (cid:48) ) if and only if C ∗ (Φ( γ )) ∼ = C ∗ (Φ( γ (cid:48) )) .In other words, unital abelian C*-algebras can be effectively classified by simple, unital AI alge-bras.Proof. By Gelfand–Naimark duality a unital abelian C*-algebra A is isomorphic to C ( P ( A )), where P ( A ) denotes the pure states of A . We therefore only need to compose three Borel maps: The maptaking the algebra A to the space of its pure states (Lemma 3.17), the map taking a compactHausdorff space X to the Bauer simplex P ( X ) (Lemma 4.8), and the map from the space ofChoquet simplexes into the set of AI-algebras that was defined in Theorem 5.1. (cid:3) A selection theorem for exact C ∗ -algebras For 2 ≤ n < ∞ , we will denote by O n the Cuntz algebra generated by n isometries s , . . . , s n satisfying (cid:80) ni =1 s i s ∗ i = 1 (see [22, 4.2].)Kirchberg’s exact embedding Theorem states that the exact separable C ∗ -algebras are preciselythose which can be embedded into O . The purpose of this section is to prove a Borel version ofthis: There is a Borel function on Γ selecting an embedding of C ∗ ( γ ) into O for each γ ∈ Γ thatcodes an exact C ∗ -algebra. In the process we will also see that the set of γ ∈ Γ such that C ∗ ( γ ) isexact forms a Borel set. Parameterizing exact C ∗ -algebras. There is a multitude of ways of parameterizing exactseparable C ∗ -algebras, which we now describe. Eventually, we will see that they are all equivalentgood standard Borel parameterizations.Define Γ Exact = { γ ∈ Γ : C ∗ ( γ ) is exact } , and let Γ Exact , u = Γ Exact ∩ Γ u denote the set of unital exact C ∗ -algebras . An alternative parame-terization of the exact separable C ∗ -algebras is given by elements of Γ( O ) = O N , equipped withthe product Borel structure, where we identify γ ∈ O N with the C ∗ -subalgebra generated by thissequence. Let Γ u ( O ) denote the set of γ ∈ Γ( O ) which code unital C ∗ -subalgebras of O .Note that a parameterization weakly equivalent to Γ( O ) is obtained by considering in the EffrosBorel space F ( O ) of closed subsets of O , the (Borel) setSA( O ) = { A ∈ F ( O ) : A is a sub-C ∗ -algebra of O } . Recall the parameterization Ξ A u of unital separable C ∗ -algebras from 2.6. We define Ξ A u , Exact to bethe subset of Ξ A u corresponding to exact unital C ∗ -algebras. Recall also that A is the free countableunnormed Q ( i )- ∗ -algebra, A u the unital counterpart. DefineˆΓ A ( O ) = { ξ : A → O : ξ is a Q ( i )- ∗ -algebra homomorphism A → O } and ˆΓ A u ( O ) = { ξ : A u → O : ξ is a unital Q ( i )- ∗ -algebra homomorphism A u → O } , and note that ˆΓ A ( O ) and ˆΓ A u ( O ) are closed (and therefore Polish) in the subspace topology, when O A and O A u are given the product topology. As previously noted, A can be identified with theset of formal Q ( i )- ∗ -polynomials p n in the formal variables X i without constant term, and A u withthe formal Q ( i )- ∗ -polynomials (allowing a constant term), which we enumerated as q n . We define g : ˆΓ A u ( O ) → Ξ A u by g ( ξ )( q n ) = (cid:107) ξ ( q n ) (cid:107) O . Note that g is continuous. By the exact embeddingTheorem we have g (ˆΓ A u ( O )) = Ξ A u , Exact .Define an equivalence relation E g in ˆΓ A u ( O ) by ξE g ξ (cid:48) ⇐⇒ g ( ξ ) = g ( ξ (cid:48) ) . For ξ ∈ ˆΓ A u ( O ), a norm is defined on A u / ker( ξ ) by letting (cid:107) q n ker( ξ ) (cid:107) ξ = (cid:107) ξ ( q n ) (cid:107) O . We define A u ( ξ ) to be the unital C ∗ -algebra obtained from completing ( A u , (cid:107) · (cid:107) ξ ), and we note that ξ extendsto an injection ¯ ξ : A u ( ξ ) → O . It is clear that the definition of A u is E g -invariant. Proposition 6.1.
With notation as above, there is a Borel set in ˆΓ A u ( O ) meeting every E g classexactly once (i.e., there is a Borel transversal for E g ). Before giving the proof, we first prove two general lemmas.
Lemma 6.2.
Let
X, Y be Polish spaces. Suppose B ⊆ X × Y is a Borel relation such that for all x ∈ X the section B x is closed (and possibly ∅ .) Then the following are equivalent:(1) The map X → F ( Y ) : x (cid:55)→ B x is Borel;(2) proj X ( B ) is Borel and there are Borel functions f n : proj X ( B ) → Y such that for all x ∈ proj X ( B ) we have f n ( x ) ∈ B x and ( f n ( x )) n ∈ N enumerates a dense sequence in B x ; The sets Γ
Exact and Γ
Exact , u are prima facie analytic, but since we will show they are Borel, the use of thelanguage of Definition 2.1 is warranted. URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 31 (3) the relation R ⊆ X × N × Q + defined by R ( x, n, ε ) ⇐⇒ ( ∃ y ∈ Y ) y ∈ B x ∧ d ( y, y n ) < ε is Borel for some (any) complete metric d inducing the topology on Y and ( y n ) n ∈ N densein Y .In particular, if any of (1)–(3) above hold, there is a Borel function F : X → Y such that F ( x ) ∈ B x for all x ∈ X , and F ( x ) depends only on B x .Proof. The equivalence of the first two is well-known, see [13, 12.13 and 12.14]. Clearly (2) implies(3) since R ( x, n, ε ) ⇐⇒ ( ∃ i ) d ( y n , f i ( x )) < ε. To see (3) = ⇒ (1), simply notice that (3) immediately implies that for all n ∈ N and ε ∈ Q + theset { x ∈ X : B x ∩ { y ∈ Y : d ( y, y n ) < ε } (cid:54) = ∅} is Borel. This shows that the inverse images under the map x (cid:55)→ B x of the sets { F ∈ F ( Y ) : F ∩ { y ∈ Y : d ( y, y n ) < ε } (cid:54) = ∅} for n ∈ N , and ε ∈ Q + } are Borel, and since the latter sets generatethe Effros Borel structure, it follows that the map X → B ( Y ) : x (cid:55)→ F x is Borel, as required.Finally, the last statement follows from (1) and the Kuratowski–Ryll-Nardzewski Theorem ([13,Theorem 12.13]). (cid:3) Lemma 6.3.
Let
X, Y and B ⊆ X × Y be as in Lemma 6.2, and suppose moreover that proj X ( B ) is Borel. Let G be a Polish group, and suppose there is a continuous G -action on Y such that thesets B x are G -invariant for all x ∈ X , and that for all ( x, y ) ∈ B we have that the G -orbit of y ∈ B x is dense in B x . Let d be a complete metric on Y and let y n be dense in Y . Then R definedas in the previous Lemma is Borel, and so in particular (1) and (2) hold for B .Proof. It is clear from the definition that R ( x, n, ε ) ⇐⇒ ( ∃ y ∈ Y ) y ∈ B x ∧ d ( y n , y ) < ε is an analytic set. To see that it is in fact Borel, fix a dense sequence g n ∈ G . Then since all G -orbits are dense in B x we also have R ( x, n, ε ) ⇐⇒ x ∈ proj X ( B ) ∧ ( ∀ y ∈ Y ) y / ∈ B x ∨ ( ∃ i ) d ( g i · y, y n ) < ε, which gives a co-analytic definition of R , so that R is Borel. (cid:3) We now turn to the proof of Proposition 6.1. Recall that if
A, B are C ∗ -algebras, B is unital, and ϕ , ϕ : A → B are ∗ -homormorphisms, we say that ϕ and ϕ are approximately unitarily equivalent if for all finite F ⊆ A and all ε > u ∈ B such that (cid:107) u ∗ ϕ ( x ) u − ϕ ( x ) (cid:107) < ε forall x ∈ F . Proof of Proposition 6.1.
Let U ( O ) denote the unitary group of O . The group U ( O ) acts con-tinuously on ˆΓ A u ( O ) by u · ξ ( q n ) = u ∗ ξ ( q n ) u = Ad u ( ξ ( q n )) , and this action preserves the equivalence classes of E g . Further, it is clear that E g is closed as asubset of ˆΓ A u ( O ) .We claim that for all ξ ∈ ˆΓ A u ( O ), the U ( O )-classes in [ ξ ] E g are dense. To see this, let ξ (cid:48) E g ξ ,and let ¯ ξ : A u ( ξ ) → O , ¯ ξ (cid:48) : A u ( ξ (cid:48) ) → O be the injections defined before Proposition 6.1. Since A u ( ξ ) = A u ( ξ (cid:48) ), it follows by [22, Theorem 6.3.8] that ¯ ξ and ¯ ξ (cid:48) are approximately unitarilyequivalent, and so we can find u ∈ U ( O ) such that u · ξ is as close to ξ (cid:48) as we like in O A .Applying Lemma 6.3 and 6.2, we get a Borel function F : ˆΓ A u ( O ) → ˆΓ A u ( O ) selecting a uniquepoint in each E g -class. Then the set F (ˆΓ A u ( O )) = { γ ∈ ˆΓ A u ( O ) : F ( γ ) = γ } is clearly a Boreltransversal. (cid:3) From Proposition 6.1 we can obtain a Borel version of Kirchberg’s exact embedding theorem.We first need a definition.
Definition 6.4.
Let A be a separable C ∗ -algebra and γ ∈ Γ. Call Ψ : N → A a code for anembedding of C ∗ ( γ ) into A if for all n, m, k ∈ N we have:(1) If p m ( γ ) + p n ( γ ) = p k ( γ ) then Ψ( m ) + Ψ( n ) = Ψ( k );(2) if p m ( γ ) = p ∗ n ( γ ) then Ψ( m ) = Ψ( n ) ∗ ;(3) if p m ( γ ) p n ( γ ) = p k ( γ ) then Ψ( m )Ψ( n ) = Ψ( k );(4) (cid:107) Ψ( m ) (cid:107) A = (cid:107) p m ( γ ) (cid:107) .It is clear that if Ψ : N → A is such a code then there is a unique ∗ -monomorphism ˆΨ : C ∗ ( γ ) → A satisfying ˆΨ( p n ( γ )) = Ψ( p n ( γ )). If A is unital with unit 1 A and C ∗ ( γ ) is unital, and Ψ furthersatisfies(5) if 1 C ∗ ( γ ) is the unit in C ∗ ( γ ) then ˆΨ(1 C ∗ ( γ ) ) = 1 A then we will call Ψ a code for a unital embedding into A . Let P A ⊆ Γ × A N be the relation P A ( γ, Ψ) ⇐⇒ Ψ is a code for an embedding into A and, assuming A is unital, let P A u ⊆ Γ u × A N be P A u ( γ, Ψ) ⇐⇒ Ψ is a code for a unital embedding into A. We note that the sections P Aγ and ( P Au ) γ are closed for all γ ∈ Γ. Theorem 6.5 (Borel Kirchberg exact embedding Theorem, unital case) . (1) The sets Γ Exact , u , Ξ A u , Exact , Γ u ( O ) and ˆΓ A u ( O ) are Borel and provide equivalent goodparameterizations of the unital separable exact C ∗ -algebras. (2) There is a Borel function f : Γ Exact , u → O N such that f ( γ ) is a code for a unital embeddingof C ∗ ( γ ) into O for all γ ∈ Γ Exact , u . In other words, the relation P O u admits a Boreluniformization.Proof. (1) Let T be a selector for E g as guaranteed by Proposition 6.1. Then g : ˆΓ A u ( O ) → Ξ A u is injective on T , and so ran( g ) = ran( g (cid:22) T ) = Ξ A u , Exact is Borel, and admits a Borel right inverse h : Ξ A u , Exact → ˆΓ A u ( O ). Since Lemma 2.4 also holds with Y = Ξ A u , there is a Borel injection˜ g : ˆΓ A u ( O ) → Ξ A u such that g ( ξ ) (cid:39) Ξ A u ˜ g ( ξ ) for all ξ ∈ ˆΓ A u ( O ), and this shows that ˆΓ A u ( O ) andΞ A u , Exact are equivalent good parameterizations.Since Γ u and Ξ A u are equivalent (Corollary 3.15), any witness to this is also a witness to thatΓ Exact , u and Ξ A u , Exact are equivalent, in particular Γ
Exact , u is also Borel. Finally, by fixing a faithfulrepresentation of O on the Hilbert space H we obtain a Borel injection of Γ u ( O ) into Γ Exact , u ( H ),while on the other hand there clearly is a natural Borel injection from ˆΓ A u ( O ) into Γ u ( O ). Thisfinishes the proof of (1).(2) Arguing exactly as in the proof of Proposition 6.1, the action of U ( O ) on the sections of P O u satisfy Lemma 6.3, since any two injective unital embeddings of C ∗ ( γ ) are approximately unitarilyequivalent for γ ∈ Γ Exact , u . (cid:3) URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 33 Since by Lemma 3.12 the map that assigns to γ ∈ Γ its unitization is Borel, we obtain:
Theorem 6.6 (Borel Kirchberg exact embedding Theorem, non-unital case) . (1) The sets Γ Exact , Γ( O ) and ˆΓ A ( O ) are Borel and provide equivalent parameterizations ofthe separable exact C ∗ -algebras. (2) There is a Borel function f : Γ Exact → O N such that f ( γ ) is a code for a embedding of C ∗ ( γ ) into O for all γ ∈ Γ Exact . In other words, the relation P O admits a Borel uniformization. Below a group action
Conjugacy of unitary operators on a separable Hilbert space cannot be reduced to isomorphismof countable structures by [12]. However, a complete classification of this relation is providedby the spectral measures. We may therefore consider a more general notion of classifiability:being reducible to an orbit equivalence relation of a Polish group action. We don’t know whetherisomorphism of separable (simple) C*-algebras is implemented by a Polish group action. (seeQuestion 9.2 and Problem 9.3). In this section we will prove that isomorphism of unital nuclearsimple separable C ∗ -algebras is indeed Borel reducible to an orbit equivalence relation induced bya Polish group action, Theorem 7.3 below. We also note a simpler fact that this also applies toisomorphism of abelian separable C*-algebras (Proposition 7.8). Before turning to the proof of thiswe briefly discuss Question 9.2 in general. Theorem 7.1.
Assume A and B are separable weakly dense subalgebras of B ( H ) . Then A ∼ = B ifand only if there is a unitary u such that uAu ∗ = B .Proof. Only the direct implication requires a proof. Let α : A → B be an isomorphism. Fix a unitvector ξ ∈ H and let ω ξ denote the vector state corresponding to ξ , ω ξ ( a ) = ( aξ | ξ ). Since A isweakly dense in B ( H ), the restriction of ω ξ to A is a pure state of A , and similarly the restriction of ω ξ to B is a pure state of B . By [17] there is an automorphism β of B such that ω ξ (cid:22) B = β ◦ α ◦ ω ξ .The isomorphism β ◦ α extends to the isomorphism between the GNS representations of A and B corresponding to the pure states ω ξ (cid:22) A and ω ξ (cid:22) B . This automorphism of B ( H ) is implementedby a unitary u as required. (cid:3) By the previous theorem, in order to give a positive answer to Question 9.2 it would suffice tohave a natural Borel space whose points are separable C*-subalgebras of B ( H ). The space Γ definedin 2.1 appears to be similar to such a space, but the following proposition, suggested to the firstauthor by Alekos Kechris, is an obstacle to the direct approach. Proposition 7.2.
On the space Γ consider the relation γ E γ (cid:48) iff C ∗ ( γ (cid:48) ) = C ∗ ( γ ) . Then thequotient Borel structure is nonstandard, even when restricted to { γ ∈ Γ : C ∗ ( γ ) is simple, unital,and nuclear } .Proof. Note that E is Borel by Lemma 2.11. It will suffice to construct a Borel map Φ : 2 N → Γsuch that x E y if and only if C ∗ (Φ( x )) = C ∗ (Φ( y )). (Here E denotes eventual equality in thespace 2 N .) We will assure that every parameter in the range of Φ corresponds to a simple nuclearalgebra. Let H j be the two-dimensional complex Hilbert space and let ζ j denote the vector (cid:0) (cid:1) in H j . Identify H with (cid:78) j ∈ N ( H j , ζ j ). For x ⊆ N let u x = (cid:79) j ∈ x (cid:18) − . (cid:19) This is a unitary operator on H . Fix a set γ j , for j ≥
1, that generates the CAR algebra A = (cid:78) j M ( C ), represented on H so that the j ’th copy of M ( C ) maps to B ( H j ). Let Φ( x ) = γ be suchthat γ = u x and γ j , for j ≥
1, are as above.Then C ∗ (Φ( x )) = C ∗ (Φ( y )) if and only if u x u ∗ y ∈ A , if and only if x ∆ y is finite. (cid:3) A reduction to an action of
Aut( O ) . Let Aut( O ) denote the automorphism group of O ,and equip Aut( O ) with the strong topology, which makes it a Polish group. We now aim to prove: Theorem 7.3.
The isomorphism relation for nuclear simple unital separable C ∗ -algebras is Borelreducible to an orbit equivalence relation induced by a Borel action of Aut( O ) on a standard Borelspace. The proof of this requires some preparation, the most substantial part being a version of Kirch-berg’s “ A ⊗ O (cid:39) O ⊗ O Theorem” for nuclear simple unital and separable A . However, we startby noting the following: Proposition 7.4.
The set { γ ∈ Γ : C ∗ ( γ ) is simple } is Borel.Proof. We use the facts that a C*-algebra A is simple if and only if for every state φ the GNSrepresentation π φ is an isometry and that the operator norm in the GNS representation reads as (cid:107) π φ ( a ) (cid:107) = sup φ ( b ∗ b ) ≤ φ ( b ∗ a ∗ ab ) for all a ∈ A .Recall from 3.5 the coding of states. We define R ⊆ Γ × C N by R ( γ, ˆ φ ) ⇐⇒ ˆ φ codes a state on C ∗ ( γ ) . Then R is easily Borel, and as noted in 3.5, the sections R γ = { ˆ φ ∈ C N : R ( γ, ˆ φ ) } are compact. For n ∈ N and ε >
0, define Q n,ε ⊆ Γ × C N by Q n,ε ( γ, ˆ φ ) ⇐⇒ ( ∀ k )( ∀ l )( ∀ m )( p k ( γ ) = p m ( γ ) ∗ p n ( γ ) ∗ p n ( γ ) p m ( γ ) ∧ p l ( γ ) = p m ( γ ) ∗ p m ( γ ) ∧ ˆ φ ( l ) ≤ ⇒ ˆ φ ( k ) + ε ≤ (cid:107) p n ( γ ) (cid:107) ) . Then Q n,ε is Borel, and the sections ( Q n,ε ) γ are closed, and therefore compact. Thus the sets S n,ε = { γ ∈ Γ : ( ∃ ˆ φ ) Q n,ε ( γ, ˆ φ ) } are Borel, by [13, Theorem 28.8]. We claim that { γ ∈ Γ : C ∗ ( γ ) is simple } = Γ \ (cid:91) n ∈ N ,ε> S n,ε . To see this, first note that if C ∗ ( γ ) is simple then the GNS representation of any state φ on C ∗ ( γ )is faithful, and so for any n ∈ N we have(7.1) sup { φ ( p m ( γ ) ∗ p n ( γ ) ∗ p n ( γ ) p m ( γ )) : φ ( p m ( γ ) p m ( γ ) ∗ ≤ } = (cid:107) p n ( γ ) (cid:107) Hence γ / ∈ S n,ε for all n ∈ N and ε >
0. On the other hand, if γ / ∈ S n,ε for all n ∈ N and ε > C ∗ ( γ ) is simple. (cid:3) A strengthening of Proposition 7.4 will be given in [6].Since Effros has shown that the class of nuclear separable C ∗ -algebras is Borel (see [14, §
5] for aproof), we now have:
URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 35 Corollary 7.5.
The set { γ ∈ Γ : C ∗ ( γ ) is simple, nuclear and unital } is Borel. A Borel version of Kirchberg’s A ⊗ O Theorem.
For A and B fixed separable C ∗ -algebras, let Hom( A, B ) = { f : A → B : f is a ∗ -homomorphism } . Then Hom(
A, B ) ⊆ L ( A, B ), the set of bouned linear maps from A to B with operator norm atmost 1, and is closed in the strong operator topology, hence is a Polish space. We let End( A ) =Hom( A, A ).Kirchberg’s A ⊗ O Theorem states that A is nuclear simple separable and unital if and only if A ⊗ O is isomorphic to O ⊗ O . The latter is itself isomorphic to O by a result of Elliott, seee.g. [22, 7.1.2 and 5.2.1]. Our next theorem is an effective version of this theorem.Let SA u ( O ) denote the standard Borel space of closed unital ∗ -subalgebras of O . Since theparameterizations Γ Exact , u and SA u ( O ) are weakly equivalent (see 6.1), it follows from Corollary 7.5that the set SA uns ( O ) = { A ∈ SA u ( O ) : A is nuclear and simple } is Borel. We will work with this parameterization of unital nuclear simple separable C ∗ -algebrasbelow. Theorem 7.6.
There is a Borel map F : SA uns ( O ) → End( O ⊗O ) such that F ( A ) is a monomor-phism of O ⊗ O onto A ⊗ O . The proof uses an approximate intertwining argument that we now describe. Let A be a simpleunital separable nuclear C ∗ -algebra, viewed as a unital subalgebra of O . Recall that for such A , the algebra A ⊗ O (and so in particular O ⊗ O ∼ = O ) has the property that every unital ∗ -endomorphism is approximately inner (see [22, 6.3.8].) Fix a ∗ -isomorphism γ : O ⊗ O → O and a summable sequence ( (cid:15) n ) of strictly positive tolerances. We will apply Elliott’s IntertwiningArgument to the a priori non-commuting diagram A ⊗ O id (cid:47) (cid:47) ι (cid:15) (cid:15) A ⊗ O id (cid:47) (cid:47) ι (cid:15) (cid:15) A ⊗ O id (cid:47) (cid:47) ι (cid:15) (cid:15) · · ·O ⊗ O id (cid:47) (cid:47) η (cid:56) (cid:56) O ⊗ O id (cid:47) (cid:47) η (cid:56) (cid:56) O ⊗ O id (cid:47) (cid:47) η (cid:58) (cid:58) · · · where ι is the tensor product of the inclusion A (cid:44) → O with the identity map on O and η is givenby a (cid:55)→ A ⊗ γ ( a ). Let us describe the procedure step-by-step, so that we may refer back to thisdescription when arguing that the intertwining can be carried out effectively.Fix a dense sequence ( x An ) in A ⊗ O and a dense sequence ( y n ) in O ⊗ O . Fix also a densesequence of unitaries u An ∈ A ⊗ O and a dense sequence of unitaries v n ∈ O ⊗ O . We assumethat u A = 1 A ⊗O and v = 1 O ⊗O . We define by recursion a sequence of finite sets F Ak ⊆ A and G Ak ⊆ O ⊗ O , sequences ( n Ak ) k ∈ N and ( m Ak ) k ∈ N of natural numbers, and homomorphisms ι k : A ⊗ O → O ⊗ O , η k : O ⊗ O → A ⊗ O subject to the following conditions: n A = 1, m A = 1, F A = G A = ∅ , ι A = ι , η A = η , and for k > We refer the reader to [22, 2.3] for a general discussion of approximate intertwining. The argument is also knownas Elliott’s Intertwining Argument. (1) F Ak = { x Ak − } ∪ F Ak − ∪ η k − ( G Ak − ).(2) G Ak = { y k − } ∪ G Ak − ∪ ι k − ( F Ak ).(3) n Ak > n Ak − is least such that if we let η Ak = Ad( u An Ak ) ◦ η then the diagram A ⊗ O id (cid:47) (cid:47) ι Ak − (cid:15) (cid:15) A ⊗ O O ⊗ O η Ak (cid:56) (cid:56) commutes up to (cid:15) k on F Ak . This is possible because any two endomorphisms of A ⊗ O are approximately unitarily equivalent and the sequence ( u An ) is dense in the unitaries of A ⊗ O .(4) m Ak > m Ak − is least such that if we let ι Ak = Ad( v m k ) ◦ ι then the diagram A ⊗ O ι Ak (cid:15) (cid:15) O ⊗ O id (cid:47) (cid:47) η Ak (cid:56) (cid:56) O ⊗ O commutes up to (cid:15) k on G Ak . This is possible because any two endomorphisms of O ⊗ O are approximately unitarily equivalent and the sequence ( v n ) is dense in the unitaries of O ⊗ O .With these definitions the diagram A ⊗ O id (cid:47) (cid:47) ι A (cid:15) (cid:15) A ⊗ O id (cid:47) (cid:47) ι A (cid:15) (cid:15) A ⊗ O id (cid:47) (cid:47) ι A (cid:15) (cid:15) · · ·O ⊗ O id (cid:47) (cid:47) η A (cid:56) (cid:56) O ⊗ O id (cid:47) (cid:47) η A (cid:56) (cid:56) O ⊗ O id (cid:47) (cid:47) η A (cid:58) (cid:58) · · · is an approximate intertwining (in the sense of [22, 2.3.1]), and so η A ∞ : O ⊗ O → A ⊗ O : η A ∞ ( b ) = lim k →∞ η k ( b )defines an isomorphism. Proof of Theorem 7.6.
Fix a dense sequence z n ∈ O . Also, let y n , v n ∈ O ⊗O be as above. By theKuratowski-Ryll-Nardzewski Selection Theorem we can find Borel maps f n : SA u ( O ) → O suchthat ( f n ( A )) n ∈ N is a dense sequence in A . Let π : N → N be a bijection with π ( n ) = ( π ( n ) , π ( n )).Associating the Q + i Q span of { f π ( n ) ( A ) ⊗ z π ( n ) | n ∈ N } to A is clearly Borel, and this span isdense; let us denote it by x An . From (the proof of) Lemma 3.13.(2), we obtain a sequence of Borelmaps Un k : SA u ( O ) → O ⊗ O such that Un k ( A ) is dense in the set of unitaries in A ⊗ O . Welet u Ak = Un k ( A ).With these definitions there are unique Borel maps A (cid:55)→ F Ak , A (cid:55)→ G Ak , A (cid:55)→ n Ak and A (cid:55)→ m Ak satisfying (1)–(4) above; in particular, if these maps have been defined for k = l −
1, and F Al isdefined, then n Al is defined as the least natural number n greater than n Ak such that( ∀ a ∈ F Ak ) (cid:107) Ad( u An u An l − · · · u An ) ◦ η ◦ Ad( v m l − · · · v m ) ◦ ι ( a ) − a (cid:107) O < ε k . URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 37 Thus the graph of A (cid:55)→ n Al is Borel. Similarly, A (cid:55)→ m Ak is seen to be Borel for all k ∈ N . But nowwe also have that the map F : SA u ( O ) → End( O ⊗ O ) : A (cid:55)→ η A ∞ is Borel, since F ( A ) = η A ∞ ⇐⇒ ( ∀ l ) F ( A )( y l ) = lim k →∞ Ad( u n k u n k − · · · u ) ◦ η ( y l )provides a Borel definition of the graph of F . (cid:3) Proof of Theorem 7.3.
The Polish group Aut( O ) acts naturally in a Borel way on SA u ( O ) by σ · A = σ ( A ). Let E be the corresponding orbit equivalence relation. We claim that isomorphismin SA uns ( O ) is Borel reducible to E .By the previous Theorem there is a Borel map g : SA uns ( O ) → SA uns ( O ) with the followingproperties: • g ( A ) ∼ = A . • For all A ∈ SA uns ( O ) there is an isomorphism G A : A ⊗ O → O under which G A ( A ⊗ O ) = g ( A ).In other words, there is an effective unital embedding of nuclear unital simple separable C ∗ -algebrasinto O with the property that the relative commutant of the image of any such algebra in O isin fact isomorphic to O . We claim that g is a Borel reduction of ∼ = SA uns ( O ) to E .Fix A, B ∈ SA uns ( O ). Clearly if g ( A ) Eg ( B ) then A ∼ = B . On the other hand, if A ∼ = B thenthere is an isomorphism ϕ : A ⊗ O → B ⊗ O which maps A ⊗ O to B ⊗ O . Thus σ = G B ◦ ϕ ◦ G − A ∈ Aut( O )satisfies σ · A = B . (cid:3) Since by Corollary 5.2 it holds that the homeomorphism relation for compact subsets of [0 , N is Borel reducible to isomorphism of nuclear simple unital AI algebras, we recover the followingunpublished result of Kechris and Solecki: Theorem 7.7 (Kechris-Solecki) . The homeomorphism relation for compact subsets of [0 , N isbelow a group action. Note that the set A = { γ ∈ Γ : C ∗ ( γ ) is abelian } is Borel since A is clearly closed in theweak operator topology in Γ. As a subspace of Γ it therefore provides a good standard Borelparameterization for abelian C*-algebras, used in the following. Proposition 7.8.
The isomorphism relation for unital abelian separable C*-algebras is Borel re-ducible to isomorphism of AI algebras, and therefore to an orbit equivalence relation induced by aPolish group action.Proof.
For γ ∈ A we have that C ∗ ( γ ) ∼ = C ( X ) where X is the pure state space of C ∗ ( γ ). The resultnow follows by Lemma 3.17 and Corollary 5.2. An alternative proof of the last claim appeals toTheorem 7.7 instead. (cid:3) Bi-embeddability of AF algebras
In this section we will show that the bi-embeddability relation of separable unital AF algebrasis not Borel-reducible to a Polish group action (Corollary 8.2). More precisely, we prove that every K σ -equivalence relation is Borel reducible to this analytic equivalence relation. (Recall that a subsetof a Polish space is K σ if it is a countable union of compact sets.) This Borel reduction is curious since bi-embeddability of separable unital UHF algebras is bi-reducible with the isomorphism ofseparable unital UHF algebras, and therefore smooth. For f and g in the Baire space N N we define f ≤ ∞ g if and only if ( ∃ m )( ∀ i ) f ( i ) ≤ g ( i ) + mf = ∞ g if and only if f ≤ ∞ g and g ≤ ∞ f This equivalence relation, also denoted E K σ , was introduced by Rosendal in [24]. Rosendal provedthat E K σ is complete for K σ equivalence relation in the sense that (i) every K σ equivalence relationis Borel reducible to it and (ii) E K σ is itself K σ . By a result of Kechris and Louveau ([15]), E K σ isnot Borel reducible to any orbit equivalence relation of a Polish group action. In particular, E K σ ,or any analytic equivalence relation that Borel-reduces E K σ , is not effectively classifiable by theElliott invariant.If A and B are C*-algebras then we denote by A (cid:44) → B the existence of a ∗ -monomorphism of A into B ; A is therefore bi-embeddable with B if and only if A (cid:44) → B and B (cid:44) → A . Proposition 8.1.
There is a Borel-measurable map N N → Γ : f (cid:55)→ C f such that(1) each C f is a unital AF algebra;(2) C f isomorphically embeds into C g if and only if f ≤ ∞ g ;(3) C f is bi-embeddable with C g if and only if f = ∞ g .Proof. We first describe the construction and then verify it is Borel. Let p i , i ∈ N , be the increasingenumeration of all primes. For f ∈ N N and n ∈ N define UHF algebras A f = ∞ (cid:79) i =1 M p f ( i ) i ( C ) and B n = ∞ (cid:79) i =1 M p ni ( C ) . Hence B n is isomorphic to A f if f ( i ) = n for all i . For f and g we have that A f (cid:44) → A g if and only if f ( i ) ≤ g ( i ) for all i . Also, A f (cid:44) → A g ⊗ B n if and only if f ( i ) ≤ g ( i ) + n for all i . Therefore, f ≤ ∞ g if and only if A f (cid:44) → A g ⊗ B n for a large enough n . Let C f be the unitization of A f ⊗ (cid:76) ∞ n =1 B n . We claim that f ≤ ∞ g if and only if C f (cid:44) → C g .First assume f ≤ ∞ g and let n be such that f ( i ) ≤ g ( i ) + n for all i . Then by the above A f ⊗ B m (cid:44) → A g ⊗ B m + n for all m , and therefore C f (cid:44) → C g .Now assume C f (cid:44) → C g . Then in particular A f (cid:44) → (cid:76) ∞ n =1 A g ⊗ B n . Since A f is simple, we have A f (cid:44) → A g ⊗ B n for some n and therefore f ≤ ∞ g . We have therefore proved that the map f (cid:55)→ C f satisfies (2). Clause (3) follows immediately.It remains to find a Borel measurable map Φ : N N → Γ such that C ∗ (Φ( f )) is isomorphic to C f for all f . Since ⊗ (for nuclear C*-algebras) and (cid:76) are Borel (by Lemma 3.7 for the former; thelatter is trivial), it suffices to show that there is a Borel map Ψ : N N → Γ such that C ∗ (Ψ( f )) isisomorphic to A f for all f .Let D denote the maximal separable UHF algebra, (cid:78) ∞ i =1 (cid:78) ∞ n =1 M p i ( C ). Let φ be its uniquetrace and let π φ : D → B ( H φ ) be the GNS representation corresponding to φ . Then H φ is a tensorproduct of finite-dimensional Hilbert spaces H n,i such that dim( H n,i ) = i . Also, for each pair n, i there is an isomorphic copy D n,i of M i ( C ) acting on H n,i and a unit vector ξ n,i such that ω ξ n,i agrees with the normalized trace on D n,i . The algebra generated by D n,i , for n, i ∈ N , is isomorphicto D .Now identify H φ with H as used to define Γ. Each D n,i is singly generated, so we can fix agenerator γ n,i . Fix a bijection χ between N and N , and write χ ( n ) = ( χ ( n ) , χ ( n )). For f ∈ N N let Ψ( f ) = γ be defined by γ n = 0 if f ( χ ( n )) < χ ( n ) and γ n = γ χ ( n ) ,χ ( n ) if f ( χ ( n )) ≥ χ ( n ). URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 39 Then C ∗ ( γ ) is isomorphic to the tensor product of the D n,i for n ≤ f ( i ), which is in turn isomorphicto A f . Moreover, the map f (cid:55)→ Ψ( f ) is continuous when Γ is considered with the product topology,because finite initial segments of Ψ( f ) are determined by finite initial segments of f . (cid:3) Corollary 8.2. E K σ is Borel reducible to the bi-embeddability relation E on separable AF C*-algebras. Therefore E is not Borel reducible to a Polish group action. (cid:3) Concluding remarks and open problems
In this section we discuss several open problems and possible directions for further investigationsrelated to the theme of this paper. The first is related to Section 8.
Problem 9.1.
Is the bi-embeddability relation for nuclear simple separable C ∗ -algebras a completeanalytic equivalence relation? What about bi-embeddability of AF algebras? We remark that the bi-embeddability, and even isomorphism, of separable Banach spaces is knownto be complete for analytic equivalence relation ([30]). Moreover, bi-embeddability of countablegraphs is already complete for analytic equivalence relations by [19]
Question 9.2.
Is isomorphism of separable (simple) C*-algebras implemented by a Polish groupaction?
George Elliott observed that the isomorphism of nuclear simple separable C*-algebras is Borelreducible to an orbit equivalence relation induced by a Borel action of the automorphism group of O ⊗ K . This is proved by an extension of the proof of Theorem 7.3 together with Borel versionof Kirchberg’s result that A ⊗ O ⊗ K is isomorphic to O ⊗ K for every nuclear simple separableC*-algebra A . The following is an extension of Question 9.2. Problem 9.3.
What is the Borel cardinality of the isomorphism relation of larger classes of sepa-rable C*-algebras (simple or not), such as:(i) nuclear C*-algebras;(ii) exact C*-algebras;(iii) arbitrary C*-algebras?Do these problems have strictly increasing Borel cardinality? Are all of them Borel reducible to theorbit equivalence relation of a Polish group action on a standard Borel space?
We can ask still more of the classes in Problem 9.3. On the space ∆ N (recall that ∆ is the closedunit disk in C ) define the relation E by letting x E y if x ( n ) = y ( n ) for all but finitely many n . In[15] it was proved that E is not Borel-reducible to any orbit equivalence relation of a Polish groupaction, and therefore E ≤ B E implies E is not Borel-reducible to an orbit equivalence relation of aPolish group action. Kechris and Louveau have even conjectured that for Borel equivalence relationsreducing E is equivalent to not being induced by a Polish group action. While Corollary 5.2 impliesthat the relations considered in Problem 9.3 are not Borel, it is natural to expect that they eitherreduce E or can be reduced to an orbit equivalence relation of a Polish group action.We defined several Borel parameterizations (see Definition 2.1) of separable C*-algebras that weresubsequently shown to be equivalent. The phenomenon that all natural Borel parameterizationsof a given classification problem seem to wind up being equivalent has been observed by otherauthors. One may ask the following general question. Problem 9.4.
Assume Γ and Γ are good standard Borel parameterizations that model isomor-phism of structures in the same category C .Find optimal assumptions that guarantee the existence of a Borel-isomorphism Φ : Γ → Γ thatpreserves the isomorphism in class C in the sense that A ∼ = B if and only if Φ( A ) ∼ = Φ( B ) . A theorem of this kind could be regarded as an analogue of an automatic continuity theorem.We have addressed many basic C*-algebra constructions here and proved that they are Borel.We have further proved that various natural subclasses of separable C*-algebras are Borel. Thereis, however, much more to consider.
Problem 9.5.
Determine whether the following C*-algebra constructions and/or subclasses areBorel:(i) the maximum tensor product, and tensor products more generally;(ii) crossed products, full and reduced;(iii) groupoid C*-algebras;(iv) Z -stable C*-algebras;(v) C*-algebras of finite or locally finite nuclear dimension;(vi) approximately subhomogeneous (ASH) algebras;(vii) the Thomsen semigroup of a C ∗ -algebra. Items (iv)–(vi) above are of particular interest to us as they are connected to Elliott’s classificationprogram (see [4]). Items (iv) and (v) are connected to the radius of comparison by the followingconjecture of Winter and the second author.
Conjecture 9.1.
Let A be a simple separable unital nuclear C*-algebra. The following are equiv-alent:(i) A has finite nuclear dimension;(ii) A is Z -stable;(iii) A has radius of comparison zero. We have shown here that simple unital nuclear separable C ∗ -algebras form a Borel set. We willshow in a forthcoming article that separable C*-algebras with radius of comparison zero also forma Borel set. Thus, those A as in the conjecture which satisfy (iii) form a Borel set. It would beinteresting to see if the same is true if one asks instead for (i) or (ii). Clearly, the Z -stable algebrasform an analytic set. As for item (vi) of Problem 9.5, the question of whether every unital simpleseparable nuclear C ∗ -algebra with a trace is ASH has been open for some time. One might tryto attack this question by asking where these formally different classes of algebras sit in the Borelhierarchy.In [3], Elliott introduced an abstract approach to functorial classification. A feature of his con-struction is that morphisms between classifying invariants lift to morphisms between the objectsto be classified. This property is shared with the classification of C*-algebras, where morphismsbetween K-theoretic invariants lift to (outer, and typically not unique) automorphisms of the orig-inal objects. It would be interesting to have a set-theoretic analysis of this phenomenon parallel tothe set-theoretic analysis of abstract classification problems used in the present paper. URBULENCE, ORBIT EQUIVALENCE, AND THE CLASSIFICATION OF NUCLEAR C ∗ -ALGEBRAS 41 Problem 9.6.
Is there a set-theoretic model for the functorial inverse in the sense of Elliott? Moreprecisely, if the categories are modelled by Borel spaces and the functor that assigns invariants isBorel, is the inverse functor necessarily Borel?
Finally, the following question was posed by Greg Hjorth to the third author:
Problem 9.7. Is (cid:39) Ell
Borel reducible to (cid:39) Λ ? That is, does isomorphism of Elliott invariants Borelreduce to affine isomorphism of separable metrizable Choquet simplexes? We will show in a forthcoming article that at least the Elliott invariant is below a group action.It would also be natural to try to obtain an answer to the following:
Problem 9.8.
Is isomorphism of separable metrizable Choquet simplexes Borel reducible to home-omorphism of compact Polish spaces? I.e., is (cid:39) Λ Borel reducible to (cid:39) K homeo ? In connection to this problem we should point out a misstatement in [10]. In [10, p. 326] it wasstated that Kechris and Solecki have proved that the homeomorphism of compact Polish spaces isBorel bi-reducible to E X ∞ G ∞ . The latter is an orbit equivalence relation of a Polish group action withthe property that every other orbit equivalence relation of a Polish group action is Borel-reducibleto it. If true, this would give a positive solution to Problem 9.8. However, Kechris and Solecki haveproved only one direction, referred to in Theorem 1.4. The other direction is still open. References
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