Rational K-Stability of Continuous C(X)-Algebras
aa r X i v : . [ m a t h . OA ] F e b RATIONAL K-STABILITY OF CONTINUOUS C ( X ) -ALGEBRAS APURVA SETH, PRAHLAD VAIDYANATHAN
Abstract.
We show that the property of being rationally K -stable passes from the fibers of a continuous C ( X )-algebra to theambient algebra, under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension. As anapplication, we show that a crossed product C*-algebra is (ratio-nally) K-stable provided the underlying C*-algebra is (rationally)K-stable, and the action has finite Rokhlin dimension with com-muting towers. Given a compact Hausdorff space X , a continuous C ( X )-algebra is thesection algebra of a continuous field of C*-algebras over X . Such alge-bras form an important class of non-simple C*-algebras, and it is oftenof interest to understand those properties of a C*-algebra which passfrom the fibers to the ambient C ( X )-algebra.Given a unital C*-algebra, we write U n ( A ) for the group of n × n unitary matrices over A . This is a topological group, and its homo-topy groups π j ( U n ( A )) are termed the nonstable K -theory groups of A . These groups were first systematically studied by Rieffel [20] in thecontext of noncommutative tori. Thomsen [26] built on this work, anddeveloped the notion of quasi-unitaries, thus constructing a homologytheory for (possibly non-unital) C*-algebras.Unfortunately, the nonstable K-theory for a given C*-algebra is noto-riously difficult to compute explicitly. Even for the algebra of complexnumbers, these groups are naturally related to the homotopy groupsof spheres π j ( S n ), which are not known for many values of j and n .It is here that rational homotopy theory has proved to be useful totopologists and, in this paper, we employ this tool in the context ofC*-algebras. Mathematics Subject Classification.
Primary 46L85; Secondary 46L80.
Key words and phrases.
Nonstable K-theory, C*-algebras.
A C*-algebra A is said to be K -stable if the homotopy groups π j ( U n ( A ))are naturally isomorphic to the K -theory groups K j +1 ( A ), and ratio-nally K-stable if the analogous statement holds for the rational homo-topy groups (see Definition 1.3). In [23], we proved that, for a contin-uous C ( X )-algebras, the property of being K -stable passes from thefibers to the whole algebra, provided the underlying space X is metriz-able and has finite covering dimension. The goal of this paper is toprove an analogous result for rational K -stability. Theorem A.
Let X be a compact metric space of finite covering di-mension and let A be a continuous C ( X ) -algebra. If each fibre of A isrationally K -stable, then so is A . As an interesting application of these results, we consider crossed prod-uct C*-algebras where the action has finite Rokhlin dimension (withcommuting towers). A theorem of Gardella, Hirshberg and Santiago [9]states that such a crossed product C*-algebra can be locally approxi-mated by a continuous C ( X )-algebra (see Definition 3.3). This leadsto the following result. Theorem B.
Let α : G → Aut ( A ) be an action of a compact Lie groupon a separable C*-algebra A such that α has finite Rokhlin dimensionwith commuting towers. If A is rationally K -stable ( K -stable), then sois A ⋊ α G . The paper is organized as follows: In Section 1, we introduce the ba-sic notions used throughout the paper - that of nonstable K -groups, C ( X )-algebras, and the rationalization of H -spaces. In Section 2, weprove Theorem A along with some applications and examples. Finally,Section 3 is devoted to the proof of Theorem B.1. Preliminaries
Nonstable K -theory. We begin by reviewing the work of Thom-sen of constructing the nonstable K -groups associated to a C*-algebra.For the proofs of the results mentioned in this section, the reader isreferred to [26].Let A be a C*-algebra (not necessarily unital). Define an associativecomposition · on A by(1) a · b = a + b − ab An element u ∈ A is said to be a quasi-unitary if u · u ∗ = u ∗ · u = 0 . ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 3 We write b U ( A ) for the set of all quasi-unitary elements in A . Forelements u, v ∈ b U ( A ), we write u ∼ v if there is a continuous function f : [0 , → b U ( A ) such that f (0) = u and f (1) = v . We write b U ( A )for the set of u ∈ b U ( A ) such that u ∼
0. Note that b U ( A ) is a closed,normal subgroup of b U ( A ). We now define the two functors we areinterested in. Definition 1.1.
Let A be a C ∗ -algebra, and k ≥ m ≥ G k ( A ) := π k ( b U ( A )) , and F m ( A ) := π m ( b U ( A )) ⊗ Q ∼ = G m ( A ) ⊗ Q . Recall [21] that a homology theory on the category of C ∗ -algebras isa sequence { h n } of covariant, homotopy invariant functors from thecategory of C ∗ -algebras to the category of abelian groups such that, if0 → J ι −→ B p −→ A → C ∗ -algebras, thenfor each n ∈ N , there exists a connecting map ∂ : h n ( A ) → h n − ( J ),making the following sequence exact . . . ∂ −→ h n ( J ) h n ( ι ) −−−→ h n ( B ) h n ( p ) −−−→ h n ( A ) ∂ −→ h n − ( J ) → . . . and furthermore, ∂ is natural with respect to morphisms of short ex-act sequences. Furthermore, we say that a homology theory { h n } iscontinuous if, whenever A = lim A i is an inductive limit in the cate-gory of C ∗ -algebras, then h n ( A ) = lim h n ( A i ) in the category of abeliangroups. The next proposition is a consequence of [26, Proposition 2.1]and [10, Theorem 4.4]. Proposition 1.2.
For each m ≥ , G m and F m are continuous homol-ogy theories. The notion of K -stability given below is due to Thomsen [26, Definition3.1], and that of rational K -stability has been studied by Farjoun andSchochet [5, Definition 1.2], where it was termed rational Bott-stability. Definition 1.3.
Let A be a C ∗ -algebra and j ≥
2. Define ι j : M j − ( A ) → M j ( A ) to be the natural inclusion map a (cid:18) a
00 0 (cid:19) .A is said to be K -stable if G k ( ι j ) : G k ( M j − ( A )) → G k ( M j ( A )) is anisomorphism for all k ≥ j ≥ A is said to be rationally K -stable if F m ( ι j ) : F m ( M j − ( A )) → F m ( M j ( A )) is an isomorphism forall m ≥ j ≥ APURVA SETH, PRAHLAD VAIDYANATHAN
Note that, for a K -stable C ∗ -algebra, G k ( A ) ∼ = K k +1 ( A ) and for arationally K -stable C ∗ -algebra, F m ( A ) ∼ = K m +1 ( A ) ⊗ Q . A varietyof interesting C*-algebras are known to be K -stable (see [23, Remark1.5]). Clearly, K -stability implies rational K -stability. By [22, Theo-rem B], the converse is true for AF-algebras. However, as Example 2.1shows, the converse is not true in general.1.2. C ( X ) -algebras. Let A be a C ∗ -algebra, and X a compact Haus-dorff space. We say that A is a C ( X )-algebra [13, Definition 1.5] if thereis a unital ∗ -homomorphism θ : C ( X ) → Z ( M ( A )), where Z ( M ( A ))denotes the center of the multiplier algebra of A . For simplicity of no-tation, if f ∈ C ( X ) and a ∈ A , we write f a := θ ( f )( a ).If Y ⊂ X is closed, the set C ( X, Y ) of functions in C ( X ) that vanishon Y is a closed ideal of C ( X ). Hence, C ( X, Y ) A is a closed, two-sided ideal of A . The quotient of A by this ideal is denoted by A ( Y ),and we write π Y : A → A ( Y ) for the quotient map (also referredto as the restriction map). If Z ⊂ Y is a closed subset of Y , wewrite π YZ : A ( Y ) → A ( Z ) for the natural restriction map, so that π Z = π YZ ◦ π Y . If Y = { x } is a singleton, we write A ( x ) for A ( { x } ) and π x for π { x } . The algebra A ( x ) is called the fiber of A at x . For a ∈ A ,write a ( x ) for π x ( a ). For each a ∈ A , we have a mapΓ a : X → R given by x
7→ k a ( x ) k . This map is, in general, upper semi-continuous [14, Lemma 2.3]. We saythat A is a continuous C ( X )-algebra if Γ a is continuous for each a ∈ A .If A is a C ( X )-algebra, we will often have reason to consider other C ( X )-algebras obtained from A . At that time, the following result ofKirchberg and Wasserman will be useful. Theorem 1.4. [14, Remark 2.6]
Let X be a compact Hausdorff space,and let A be a continuous C ( X ) -algebra. If B is a nuclear C ∗ -algebra,then A ⊗ B is a continuous C ( X ) -algebra whose fiber at a point x ∈ X is A ( x ) ⊗ B . In particular, if A is a continuous C ( X )-algebra, then so is M ( A ).If Y ⊂ X is a closed set, we will denote the restriction map by η Y : M ( A ) → M ( A ( Y )), and we write ι Y : A ( Y ) → M ( A ( Y )) for thenatural inclusion map. If Y = X , we simply write ι (or ι A ) for ι X . Notethat η Y ◦ ι = ι Y ◦ π Y . Once again, if Y = { x } , we simply write ι x for ι { x } .Finally, the notion of a pullback is important for our investigation: Let B, C, and D be C ∗ -algebras, and δ : B → D and γ : C → D be ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 5 ∗ -homomorphisms. We define the pullback of this system to be A = B ⊕ D C := { ( b, c ) ∈ B ⊕ C : δ ( b ) = γ ( c ) } . This is described by a diagram(2) A φ / / ψ (cid:15) (cid:15) B δ (cid:15) (cid:15) C γ / / D where φ ( b, c ) = b and ψ ( b, c ) = c . The next lemma allows us to induc-tively put together a C ( X )-algebra from its natural quotients. Lemma 1.5. [2, Lemma 2.4]
Let X be a compact Hausdorff space and Y and Z be two closed subsets of X such that X = Y ∪ Z . If A is a C ( X ) -algebra, then A is isomorphic to the pullback A π Y / / π Z (cid:15) (cid:15) A ( Y ) π YY ∩ Z (cid:15) (cid:15) A ( Z ) π ZY ∩ Z / / A ( Y ∩ Z ) . Rational Homotopy Theory.
We now discuss some basic factsabout the rationalization of groups and spaces as developed in [11].A connected CW-complex Y is said to be nilpotent if π ( Y ) is a nilpo-tent group and π ( Y ) acts nilpotently on π j ( Y ) for all j ≥
2. A nilpo-tent space Y is a rational space if, for each j ≥
1, the homotopy group π j ( Y ) is a Q -vector space. A continuous map r : Y → Z is said to be arationalization of Y if Z is a rational space and r ∗ ⊗ id : π ∗ ( Y ) ⊗ Q → π ∗ ( Z ) ⊗ Q ∼ = π ∗ ( Z ) is an isomorphism. The next theorem (see [11,Theorem II.3A]) is fundamental to the theory. Theorem 1.6 (Hilton, Mislin, and Roitberg) . Every nilpotent CWcomplex Y has a rationalization r : Y → Y Q , where Y Q is a CW com-plex. The space Y Q is uniquely determined up to homotopy equivalence. We now specialize to the situation of our interest. Recall that an H -space is a pointed space ( Y, e ) endowed with a ‘multiplication’ map µ : Y × Y → Y such that e is a homotopy unit, that is, the maps λ, ρ : Y → Y given by λ ( y ) := µ ( e, y ) and ρ ( y ) := µ ( y, e ) are bothhomotopic to id Y . We denote this by the triple ( Y, e, µ ). We saythat (
Y, e, µ ) is homotopy-associative if the maps µ ◦ ( µ × id Y ) and µ ◦ (id Y × µ ) : Y × Y × Y → Y are homotopic. In what follows, wewill implicitly assume that the H -spaces under consideration are all APURVA SETH, PRAHLAD VAIDYANATHAN homotopy-associative.Now suppose (
Y, e, µ ) is an H -space, where the space Y is a connectedCW-complex. Since Y is nilpotent, it has a rationalization r : Y → Y Q by Theorem 1.6. Now, by [17, Theorem 6.2.3], r × r : Y × Y → Y Q × Y Q is a rationalization. By the universal property of the rationalization,there is a unique map ρ : Y Q × Y Q → Y Q such that the following diagramcommutes upto homotopy.(3) Y × Y µ / / r × r (cid:15) (cid:15) Y r (cid:15) (cid:15) Y Q × Y Q ρ / / Y Q . By the mapping cylinder construction, we may assume that r is acofibration. Then, r × r is also a cofibration as it is the composition oftwo cofibrations Y × Y → Y × Y Q → Y Q × Y Q . Hence, by [24, Problem5.3], we may assume that the above diagram commutes strictly. Ifwe set e Q := r ( e ), then it follows from [17, Proposition 6.6.2] thatthe triple ( Y Q , e Q , ρ ) is an H -space. Furthermore, by universality, wemay also ensure that the triple ( Y, e Q , ρ ) is homotopy-associative. Wesummarize this result below. Proposition 1.7. If ( Y, e, µ ) is a homotopy-associative H -space, where Y is a connected CW-complex, then there is a homotopy-associative H -space ( Y Q , e Q , ρ ) and a map r : Y → Y Q such that r is a rationalization,and the diagram Eq. (3) commutes strictly. If A is a C*-algebra, then b U ( A ) has the homotopy type of a CW-complex [26, Corollary 1.6]. Therefore, b U ( A ) may be regarded as aconnected CW-complex. Since b U ( A ) is a topological group (and hencea connected H -space), it has as rationalization r : b U ( A ) → b U ( A ) Q .By Proposition 1.7, b U ( A ) Q has the structure of an H -space, which wewrite as ( b U ( A ) Q , e Q , ρ ), where e Q = r (0). Finally, observe that thecommutativity of Eq. (3) implies that ρ ( e Q , e Q ) = e Q .1.4. Notational Conventions. If A and B are two C ∗ -algebras, thesymbol A ⊗ B will always denote the minimal tensor product. If B = C ( X ) is commutative, we identify C ( X ) ⊗ A with C ( X, A ),the space of continuous A -valued functions on X that vanish at infinity.Suppose f and g are two continuous paths in a topological space Y . If f (1) = g (0), we write f • g for the concatenation of the two paths. If f ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 7 and g agree at end-points, we write f ∼ h g if there is a path homotopybetween them. Furthermore, we write f for the path f ( t ) := f (1 − t )and the constant path at a point ∗ as e ∗ .If X and Y are two pointed spaces, we write C ∗ ( X, Y ) for the space ofbase point preserving continuous functions from X to Y . Note that if A is a C*-algebra, and Y is either A or b U ( A ), then we always take 0to be the base point. In that case, C ∗ ( X, A ) is a C*-algebra, and, forany path-connected space X , there is a natural isomorphism b U ( C ∗ ( X, A )) ∼ = C ∗ ( X, b U ( A )) . Henceforth, we will identify these two spaces without further comment.If (
Y, e, µ ) is an H -space and a ∈ Y , we may define non-negative powersof a inductively by µ ( a ) := e and µ n ( a ) := µ ( µ n − ( a ) , a ). Similarly,if f : X → Y is any function, we define non-negative powers of f point-wise, that is, µ n ( f )( x ) := µ n ( f ( x )) for all n ≥
0. Note that,if f ∈ C ∗ ( S j , Y ), then [ µ n ( f )] = n [ f ] in π j ( Y ) by [28, Theorem 4.7].Throughout the rest of the paper, for any C ∗ -algebra B , we write µ B for the multiplication in b U ( B ) given by Eq. (1), and ρ B for themultiplication in b U ( B ) Q given by Proposition 1.7.2. Main Results
The goal of this section is to provide a proof for Theorem A. To putthings in perspective, we begin by constructing an example of a C*-algebra that is rationally K -stable, but not K -stable. Example 2.1.
Let X be a connected, finite CW-complex such that H i ( X ; Z ) is a finite group for all i ≥ X tobe the real projective space RP ), and set A := C ∗ ( X, C ) . Note that, for all n, m ≥ F n ( M m ( A )) = π n ( C ∗ ( X ; U m )) ⊗ Q = M l ≥ n ˜ H l − n ( X ; π l ( U m ) ⊗ Q ) = 0by [15, Theorem 4.20]. Hence, A is rationally K -stable.Now suppose that A is K -stable. We fix a path connected H -space Y , and consider the following fibration sequence (see the proof of [15,Proposition 4.9]) C ∗ ( X, Y ) → C ( X, Y ) → Y. APURVA SETH, PRAHLAD VAIDYANATHAN
This fibration has a section, hence the long exact homotopy sequencebreaks into split short exact sequences(4) 0 → π n ( C ∗ ( X, Y )) → π n ( C ( X, Y )) → π n ( Y ) → n ∈ N . By a result of Thom [25, Theorem 2], if Y = S = K ( Z , π n ( C ( X, S )) ∼ = H − n ( X ; Z ). It follows that G n ( A ) = π n ( b U ( C ∗ ( X, C ))) ∼ = π n ( C ∗ ( X, S )) = 0for all n ≥
1. If A were K -stable, it would follow that π n ( C ∗ ( X, U m )) ∼ = G n ( M m ( A )) ∼ = G n ( A ) = 0for all n, m ≥
1. Hence, we conclude that π n ( C ∗ ( X, b U ( K ))) ∼ = G n ( A ⊗K ) = 0 for all n ≥
1. Taking Y = b U ( K ) in Eq. (4), we conclude that(5) π n ( C ( X, b U ( K ))) ∼ = π n ( b U ( K )) = ( Z : n odd0 : n even . Thus, in order to show that A is not K -stable, it suffices to showthat Eq. (5) cannot hold. To do this, we consider the work of Federer[6], who constructed a spectral sequence converging to these homotopygroups (note that X is a finite CW-complex, and b U ( K ) is a simplespace, so the results of [6] do apply). The first page of this spectralsequence, which converges to π p ( C ( X, b U ( K ))), is of the form C (1) p,q ∼ = H q ( X ; π p + q ( b U ( K )))with differential d : C (1) p,q → C (1) p − ,q +2 . Therefore, for C (1) p,q to be non-zero, p + q must be odd. But in that case, C (1) p − ,q +2 is zero. Hence,the spectral sequence collapses at the very first page, so C (1) p,q = C ( ∞ ) p,q .Therefore, we conclude that π n ( C ( X, b U ( K ))) = M q ≥ H q ( X ; π n + q ( b U ( K )))for all n ≥
1. This is a finite sum of finite groups (by our choice of X ),contradicting Eq. (5). Thus, A is not K -stable.We now turn to the proof of Theorem A, and begin with some lemmasthat will be useful to us. The first lemma, which we will use repeatedlythroughout the paper, follows from [26, Theorem 1.9] and [3, Theorem4.8]. Lemma 2.2.
Let ϕ : A → B be a surjective ∗ -homomorphism betweentwo C*-algebras, then the induced maps ϕ : b U ( A ) → ϕ ( b U ( A )) and ϕ : b U ( A ) → b U ( B ) are both Serre fibrations. ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 9 Lemma 2.3. [23, Lemma 2.2]
Let a, b ∈ b U ( A ) such that k a − b k < ,then a ∼ b in b U ( A ) . Note that, for any element a in a C*-algebra A (not necessarily a quasi-unitary), we write µ AN ( a ) for a · a · . . . a ( N times). The next lemma isa variation of [23, Lemma 2.3] that we need for our purposes. Lemma 2.4.
For any ǫ > and any N ∈ N , there exists δ > satis-fying the following condition: For any C*-algebra A , and any element a ∈ A such that k a k ≤ , k a · a ∗ k < δ , and k a ∗ · a k < δ , there exists aquasi-unitary u ∈ b U ( A ) such that k µ AN ( u ) − µ AN ( a ) k < ǫ. Proof.
Note that the function d µ AN ( d ) is a polynomial in d (that isindependent of A ). Thus, for any ǫ >
0, there exists η > A and any c, d ∈ A with k c k , k d k ≤ k c − d k < η , we have k µ AN ( c ) − µ AN ( d ) k < ǫ .We choose δ > ǫ = η , then there exists u ∈ b U ( A ) such that k u − a k < η , so that k µ AN ( u ) − µ AN ( a ) k < ǫ . (cid:3) Our proof of Theorem A is by induction on the covering dimension ofthe underlying space. The next theorem is the base case, and it holdseven if the space is not metrizable. In what follows, we will repeatedlyuse the fact that, for any abelian group A , any element in A ⊗ Q canbe represented as an elementary tensor of the form u ⊗ /m for some u ∈ A and m ∈ Z . Theorem 2.5.
Let X be a compact Hausdorff space of zero coveringdimension, and let A be a continuous C ( X ) -algebra. If each fiber of A is rationally K -stable, then so is A .Proof. We show that the map ι ∗ ⊗ id : π j ( b U ( A )) ⊗ Q → π j ( b U ( M n ( A ))) ⊗ Q is an isomorphism for each n ≥ j ≥
1. For simplicity of notation,we fix n = 2.We first consider injectivity. Suppose [ f ] ⊗ q ∈ π j ( b U ( A )) ⊗ Q such that[ ι ◦ f ] ⊗ q = 0 in π j ( b U ( M ( A )) ⊗ Q . Then by elementary group theory[ ι ◦ f ] has finite order in π j ( b U ( M ( A ))). Thus for x ∈ X , [ ι x ◦ π x ◦ f ]has finite order in π j ( b U ( M ( A ( x ))). Since A ( x ) is rationally K -stable, [ π x ◦ f ] has finite order in π j ( b U ( A ( x ))). Hence, there exists N x ∈ N and a path F : [0 , → C ∗ ( S j , b U ( A ( x ))) such that F (0) = 0 , and F (1) = µ A ( x ) N x ( π x ◦ f ) . By [23, Lemma 2.4], there is a closed neighbourhood Y x of x suchthat µ A ( Y x ) N x ( π Y x ◦ f ) ∼ C ∗ ( S j , b U ( A ( Y x ))). Since X is zero di-mensional, we may assume that the sets { Y x : x ∈ X } are cl-openand disjoint. Since X is compact, we may obtain a finite sub-cover { Y x , Y x , . . . , Y x n } . By Lemma 1.5, A ∼ = A ( Y x ) ⊕ A ( Y x ) ⊕ . . . ⊕ A ( Y x n )via the map b ( π Y x ( b ) , π Y x ( b ) , . . . , π Y xn ( b )). If N := lcm ≤ i ≤ n ( N x i ),then µ A ( Y xi ) N ( π Y xi ◦ f ) ∼ C ∗ ( S j , b U ( A ( Y x i ))), for each 1 ≤ i ≤ n .Thus, µ N ( f ) ∼ π j ( b U ( A )). Hence, [ f ] has finite order in π j ( b U ( A )),so [ f ] ⊗ q = 0 in π j ( b U ( A )) ⊗ Q . Thus, ι ∗ ⊗ id is injective.For surjectivity, choose [ u ] ∈ π j ( b U ( M ( A ))) and m ∈ Z non-zero. Wewish to construct an element [ ω ] ∈ π j ( b U ( A )) and q ∈ Q such that ι ∗ ⊗ id([ ω ] ⊗ q ) = [ u ] ⊗ m . To this end, fix x ∈ X . Since A ( x ) is rationally K -stable, there exists[ f x ] ∈ π j ( b U ( A ( x ))) and q x ∈ Q such that( ι x ) ∗ ⊗ id([ f x ] ⊗ q x ) = [ η x ◦ u ] ⊗ m . Replacing f x by a multiple of itself if need be, we obtain integers L x , N x ∈ N such that N x [ ι x ◦ f x ] = L x [ η x ◦ u ]in π j ( b U ( M ( A ( x )))). Hence, there is a path g x : [0 , → C ∗ ( S j , b U ( M ( A ( x ))))such that g x (0) = µ M ( A ( x )) L x ( η x ◦ u ) and g x (1) = µ M ( A ( x )) N x ( ι x ◦ f x ). Choose e x ∈ C ∗ ( S j , A ) such that π x ◦ e x = f x . Note that e x may not be a quasi-unitary, but we may ensure that k e x k = k f x k ≤
2. Since the map η x : C ∗ ( S j , b U ( M ( A ))) → η x ( C ∗ ( S j , b U ( M ( A ))))is a fibration, g x lifts to a path G x : [0 , → C ∗ ( S j , b U ( M ( A ))) suchthat G x (0) = µ M ( A ) L x ( u ). Let b x := G x (1), and so that η x ◦ b x = µ M ( A ( x )) N x ( ι x ◦ π x ◦ e x ). Choose δ > ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 11 holds for ǫ = 1 and N = N x . Since A is a continuous C ( X )-algebra,there is a closed neighbourhood Y x of x such that k π Y x ◦ ( e ∗ x · e x ) k < δ, k π Y x ◦ ( e x · e ∗ x ) k < δ and k η Y x ◦ b x − µ M ( A ( Y x )) N x ( η Y x ◦ ι ◦ e x ) k <
1. By Lemma 2.4, thereis a quasi-unitary d x ∈ C ∗ ( S j , b U ( A ( Y x ))) such that k µ A ( Y x ) N x ( d x ) − µ A ( Y x ) N x ( π Y x ◦ e x ) k <
1, so that k µ M ( A ( Y x )) N x ( ι Y x ◦ d x ) − η Y x ◦ b x k < . By Lemma 2.3, µ M ( A ( Y x )) N x ( ι Y x ◦ d x ) ∼ η Y x ◦ b x in C ∗ ( S j , b U ( M ( A ( Y x ))).Hence, ι Y x ◦ µ A ( Y x ) N x ( d x ) ∼ µ M ( A ( Y x )) L x ( η Y x ◦ u ). As before, since X iscompact and zero-dimensional, we may choose a finite refinement of { Y x : x ∈ X } consisting of disjoint cl-open sets, which we denote by { Y x , Y x , . . . , Y x n } . Then, by Lemma 1.5, A ∼ = A ( Y x ) ⊕ A ( Y x ) ⊕ . . . ⊕ A ( Y x n )via the map a ( π Y x ( a ) , π Y x ( a ) , . . . , π Y xn ( a )). Similarly, M ( A ) ∼ = M ( A ( Y x )) ⊕ M ( A ( Y x )) ⊕ . . . ⊕ M ( A ( Y x n ))via the map b ( η Y x ( b ) , η Y x ( b ) , . . . , η Y xn ( b )). Define L := lcm ≤ i ≤ n ( L x i ),so that ι Y xi ◦ c x i ∼ µ M ( A ( Y xi )) L ( η Y xi ◦ u )in C ∗ ( S j , b U ( M ( A ( Y x i ))), where c x i ∈ C ∗ ( S j , b U ( A ( Y x i ))) is an appro-priate power of d x i . Choose ω ∈ C ∗ ( S j , b U ( A )) such that π Y xi ◦ ω = c x i for all 1 ≤ i ≤ n . Furthermore, for each 1 ≤ i ≤ n , η Y xi ◦ ι ◦ ω = ι Y xi ◦ c x i ∼ µ M ( A ( Y xi )) L ( η Y xi ◦ u )in C ∗ ( S j , b U ( M ( A ( Y x i ))), so that ι ◦ ω ∼ µ M ( A ) L ( u ) in C ∗ ( S j , b U ( M ( A ))).Thus, ι ∗ ⊗ id (cid:18) [ ω ] ⊗ Lm (cid:19) = [ u ] ⊗ m . This proves the surjectivity of ι ∗ ⊗ id. (cid:3) The next few lemmas allow us to extend this argument to higher di-mensional spaces.
Lemma 2.6.
Let ( Y, e, µ ) be an H-space, where Y is a connected CW-complex. Let r : ( Y, e, µ ) → ( Y Q , e Q , ρ ) be the rationalization map fromProposition 1.7, and let j ≥ be a fixed integer. (1) Let [ f ] ∈ π j ( Y ) and n ∈ N , and suppose there is a path H :[0 , → C ∗ ( S j , Y ) such that H (0) = e and H (1) = µ n ( f ) .Then, there exists a path G : [0 , → C ∗ ( S j , Y Q ) with G (0) = e Q and G (1) = r ◦ f , and such that r ◦ H ∼ h ρ n ( G ) in C ∗ ( S j , Y Q ) . (2) Let [ f ] ∈ π j ( Y ) , and suppose there is a path G ′ : [0 , → C ∗ ( S j , Y Q ) such that G ′ (0) = e Q and G ′ (1) = r ◦ f . Then,there exists a natural number N ∈ N and a path H ′ : [0 , → C ∗ ( S j , Y ) with H ′ (0) = e and H ′ (1) = µ N ( f ) , such that r ◦ H ′ ∼ h ρ N ( G ′ ) in C ∗ ( S j , Y Q ) .Proof. (1) Since Y Q is a rational space, [ r ◦ f ] = 0 in π j ( Y Q ). Let L : [0 , → C ∗ ( S j , Y Q ) be such that L (0) = e Q and L (1) = r ◦ f . Thus, ρ n ( L ) : [0 , → C ∗ ( S j , Y Q ) is a path that satisfies ρ n ( L )(0) = e Q and ρ n ( L )(1) = ρ n ( r ◦ f ). Note that π ( C ∗ ( S j , Y Q )) is itself a Q -vector space [11, Theorem II.3.11] and ( r ◦ H ) • ρ n ( L ) is aloop in C ∗ ( S j , Y Q ). Thus, there exists [ T ] ∈ π ( C ∗ ( S j , Y Q )) suchthat [( r ◦ H ) • ρ n ( L )] = n [ T ] = [ ρ n ( T )] . Hence, G := T • L is the required homotopy (since the operation ρ n respects concatenation).(2) Since [ r ◦ f ] = 0, in π j ( Y Q ), under r ∗ ⊗ Q : π j ( Y ) ⊗ Q → π j ( Y Q )we get that [ r ◦ f ] ∼ = [ f ] ⊗ π j ( Y ) ⊗ Q . Hence byelementary group theory, this implies that [ f ] has finite orderin π j ( Y ). Thus, there exists n ∈ N such that n [ f ] = 0 in π j ( Y )say by homotopy K : [0 , → C ∗ ( S j , Y ) such that K (0) = e, K (1) = µ n ( f ) . Now, by a similar argument to that of part (1), ( r ◦ K ) • ρ n ( G ′ )is a loop in C ∗ ( S j , Y Q ), which is a rational space. Hence, thereexists [ T ] ∈ π ( C ∗ ( S j , Y Q )) satisfying n [ T ] = [ ρ n ( T )] = [ r ◦ K • ρ n ( G ′ )] . Now n [ T ] ∈ π ( C ∗ ( S j , Y Q )) ∼ = π ( C ∗ ( S j , Y )) ⊗ Q , so there exists[ h ] ∈ π ( C ∗ ( S j , Y )) and m ∈ Z such that n [ T ] = [ r ◦ h ] m . ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 13 Thus, by the fact that ρ is homotopy-associative, m [ r ◦ K • ρ n ( G ′ )] = [ ρ m ( r ◦ K ) • ρ mn ( G ′ )] = mn [ T ] = [ r ◦ h ] . Thus, if H ′ := h • µ m ( K ) and N := mn , then H ′ (0) = e, H ′ (1) = µ N ( f ), and r ◦ H ′ is path homotopic to ρ N ( G ′ ). (cid:3) The next result will be useful to us in the following context: Suppose
B, C, and D be C ∗ -algebras, and δ : B → D and γ : C → D are ∗ -homomorphisms. Let A = B ⊕ D C be the pullback as in Eq. (2).Then, b U ( A ) may be described as a pullback (in the category of pointedtopological spaces) by the induced diagram b U ( A ) φ / / ψ (cid:15) (cid:15) b U ( B ) δ (cid:15) (cid:15) b U ( C ) γ / / b U ( D )In other words, a pair ( b, c ) ∈ A is in b U ( A ) if and only if b ∈ b U ( B ) and c ∈ b U ( C ). We now introduce some notation we will use in the future:Given a path G : [0 , → Y in a topological space Y , e G is a path givenby(6) e G ( s ) = e G (0) (3 s ) : for 0 ≤ x ≤ G (3 s −
1) : for ≤ x ≤ e G (1) (3 s −
2) : for ≤ x ≤ Lemma 2.7.
Consider a pullback diagram of pointed topological spacesgiven by P φ / / φ (cid:15) (cid:15) X π (cid:15) (cid:15) Y π / / Z such that one of the maps π or π is a Serre fibration. Let p = ( x, y ) , p ′ = ( x ′ , y ′ ) be in P , such that there exists paths G : [0 , → X, G : [0 , → Y with the property that G (0) = x , G (1) = x ′ , G (0) = y , G (1) = y ′ and π ◦ G ∼ h π ◦ G in Z . Then, there is a path H : [0 , → P suchthat H (0) = p and H (1) = p ′ . Proof.
Assume without loss of generality that π is a Serre fibration.Then since, π ◦ G ∼ h π ◦ G , there is a homotopy F : [0 , × [0 , → D such that F ( s,
0) = π ◦ G ( s ) , F ( s,
1) = π ◦ G ( s ) F (0 , t ) = π ( x ) = π ( y ) , F (1 , t ) = π ( x ′ ) = π ( y ′ ) . Then, F lifts to a homotopy F ′ : [0 , × [0 , → X , such that F ′ ( s,
0) = G , π ◦ F ′ = F, π ◦ F ′ ( t,
1) = π ◦ G ( t ) . So if we define G X ( s ) = F ′ (0 , s ) , for 0 ≤ s ≤ F ′ (3 s − , , for ≤ s ≤ F ′ (1 , − s ) , for ≤ s ≤ π ◦ G X = π ◦ f G . Therefore, the pair ( G X , f G ) defines a path in P from p to p ′ . (cid:3) Lemma 2.8.
Let X and Y be two connected topological spaces, and i : X → Y and q : Y → X be homotopy inverses of each other. For x ∈ X , let H : [0 , → Y be a path in Y , such that H (0) = i ( x ) , H (1) = i ◦ q ◦ i ( x ) Then, there exists a path T : [0 , → X such that T (0) = x, T (1) = q ◦ i ( x ) and i ◦ T is path homotopic to H in Y .Proof. Since q ◦ i ∼ h id X , there is a path S : [0 , → X such that S (0) = q ◦ i ( x ), and S (1) = x . Thus, H • ( i ◦ S ) is a loop in Y basedat i ( x ). Since π ( Y ) = i ∗ ( π ( X )), there exists a loop L based at x in X such that i ∗ [ L ] = [ H • ( i ◦ S )] . Then, T := L • S is the required path. (cid:3) Note that, if B is a C*-algebra, then the rationalization b U ( B ) Q of b U ( B ) carries an H -space structure by Proposition 1.7. We shall use ρ B to denote this multiplication map. Furthermore, we write e Q and e Q for the units of b U ( B ) Q and b U ( M ( B )) Q respectively. Proposition 2.9.
Let B be a rationally K -stable C*-algebra, [ f ] ∈ π j ( b U ( B )) and n ∈ N such that [ ι ◦ f ] is is an element of order n in π j ( b U ( M ( B )) . Let H : [0 , → C ∗ ( S j , b U ( M ( B ))) be a path satisfying H (0) = 0 and H (1) = µ M ( B ) n ( ι ◦ f ) = ι ◦ µ Bn ( f ) . ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 15 Then, there exists a natural number N ∈ N and a path H ′ : [0 , → C ∗ ( S j , b U ( B )) such that µ M ( B ) N ( H ) ∼ h ι ◦ H ′ in C ∗ ( S j , b U ( M ( B ))) .Proof. Since B is rationally K -stable, there are maps b U ( B ) Q → b U ( M ( B )) Q and b U ( M ( B )) Q → b U ( B ) Q which are homotopy inverses of each other.Therefore, we get a commuting diagram C ∗ ( S j , b U ( B )) ι / / r (cid:15) (cid:15) C ∗ ( S j , b U ( M ( B )) R (cid:15) (cid:15) C ∗ ( S j , b U ( B ) Q ) i / / C ∗ ( S j , b U ( M ( B )) Q )where r and R represent the rationalization maps. Furthermore, i has a homotopy inverse q : C ∗ ( S j , b U ( M ( B )) Q ) → C ∗ ( S j , b U ( B ) Q ).Let H : [0 , → C ∗ ( S j , b U ( M ( B ))) as above. Since R is a rational-ization map, applying Lemma 2.6, we get a homotopy G : [0 , → C ∗ ( S j , b U ( M ( B )) Q ) such that G (0) = e Q , and G (1) = R ◦ ι ◦ f = ι ◦ r ◦ f Furthermore, ρ M ( B ) n ( G ) is path homotopic to R ◦ H in C ∗ ( S j , b U ( M ( B )) Q ).Now, q ◦ G : [0 , → C ∗ ( S j , b U ( B ) Q ) is such that q ◦ G (0) = e Q , and q ◦ G (1) = q ◦ i ◦ r ◦ f Note that i and q are homotopy equivalences, hence G and i ◦ q ◦ G are homotopic in C ∗ ( S j , b U ( M ( B )) Q ) say by K : [0 , × [0 , → C ∗ ( S j , b U ( M ( B )) Q ) satisfying K ( s,
0) = G ( s ) , K ( s,
1) = i ◦ q ◦ G ( s ) , K (0 , t ) = e Q Define T : [0 , → C ∗ ( S j , b U ( M ( B )) Q ) as T ( t ) = K (1 , − t ). Then T (0) = i ◦ q ◦ i ◦ r ◦ f, T (1) = i ◦ r ◦ f Thus, by Lemma 2.8, there is a homotopy S : [0 , → C ∗ ( S j , b U ( B ) Q )such that S (0) = q ◦ i ◦ r ◦ f, S (1) = r ◦ f and i ◦ S is path homotopic to T in C ∗ ( S j , b U ( M ( B )) Q ). Since ( i ◦ q ◦ G ) • T is path homotopic to G , this implies ( i ◦ q ◦ G ) • ( i ◦ S ) is path homotopic to G in C ∗ ( S j , b U ( M ( B )) Q ). Thus, we get a path( q ◦ G ) • S : [0 , → C ∗ ( S j , b U ( B )) Q ) so that( q ◦ G ) • S (0) = e Q , q ◦ G • S (1) = r ◦ f, i ◦ ( q ◦ G • S ) ∼ h G Again, since r is a rationalization map, by Lemma 2.6, there exists anatural number m ∈ N and a path H ′ : [0 , → C ∗ ( S j , b U ( B )) suchthat H ′ (0) = 0 , H ′ (1) = µ Bm ( f ) , r ◦ H ′ ∼ h ρ Bm (( q ◦ G ) • S ) . Take k = lcm { n, m } , and write k = nℓ = mℓ for some ℓ , ℓ ∈ N .Then µ Bl ( H ′ ) : [0 , → C ∗ ( S j , b U ( B )) is such that µ Bℓ ( H ′ )(0) = 0 , µ Bℓ ( H ′ )(1) = µ Bk ( f ) , and r ◦ µ Bℓ ( H ′ ) ∼ h ρ Bk (( q ◦ G ) • S ) . Also µ M ( B ) ℓ ( H ) : [0 , → C ∗ ( S j , b U ( M ( B )) is such that µ M ( B ) ℓ ( H )(0) = 0 , µ M ( B ) ℓ ( H )(1) = ι ◦ µ Bk ( f ) , and R ◦ µ M ( B ) ℓ ( H ) ∼ h ρ M ( B ) k ( G ) . Then, from the earlier arguments, we have the following relations ρ M ( B ) k ( G ) ∼ h R ◦ µ M ( B ) ℓ ( H ) ,i ◦ (( q ◦ G ) • S ) ∼ h G, and r ◦ µ Bℓ ( H ′ ) ∼ h ρ Bk (( q ◦ G ) • S )Also i ◦ r ◦ H ′ = R ◦ ι ◦ H ′ . Hence R ◦ ι ◦ µ Bℓ ( H ′ ) = i ◦ r ◦ µ Bℓ ( H ′ ) ∼ h ρ M ( B ) k ( i ◦ (( q ◦ G ) • S )) ∼ h ρ M ( B ) k ( G ) ∼ h R ◦ µ M ( B ) ℓ ( H )Thus h R ◦ (cid:16) ι ◦ µ Bℓ ( H ′ ) • µ M ( B ) ℓ ( H ) (cid:17)i = 0in π ( C ∗ ( S j , b U ( M ( B )) Q ). Then, by Lemma 2.6, there exists a naturalnumber P ∈ N such that ι ◦ µ BP ℓ ( H ′ ) = µ M ( B ) P ( ι ◦ µ Bℓ ( H ′ )) ∼ h µ M ( B ) P ( µ M ( B ) ℓ ( H )) = µ M ( B ) P ℓ ( H )in C ∗ ( S , b U ( M ( B ))). Thus, replacing H ′ by µ BP ℓ ( H ′ ) and taking N := P ℓ , we have ι ◦ H ′ ∼ h µ M ( B ) N ( H ) . (cid:3) The next lemma is an analogue of [23, Lemma 2.7], and is a consequenceof that result and Proposition 2.9.
ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 17 Lemma 2.10.
Given X a compact Hausdorff space, A a continuous C ( X ) -algebra, and x ∈ X such that A ( x ) is rationally K -stable. For [ f ] ∈ π j ( b U ( A )) , let F : [0 , → C ∗ ( S j , b U ( M ( A )) ) be a path and n ∈ N such that F (0) = 0 and F (1) = µ M ( A ) n ( ι ◦ f ) . Then, there is a closed neighbourhood Y of x , a natural number N x ∈ N and a path L Y : [0 , → C ∗ ( S j , b U ( A ( Y )) ) such that L Y (0) =0 , L Y (1) = µ A ( Y ) N x n ( π Y ◦ f ) , and ι Y ◦ L Y ∼ h µ M ( A ( Y )) N x ( η Y ◦ F ) in C ∗ ( S j , b U ( M ( A ( Y )))) . Remark 2.11.
We are now in a position to prove Theorem A, butfirst, we need one important fact, which allows us to use induction:If X is a finite dimensional compact metric space, then covering di-mension agrees with the small inductive dimension [4, Theorem 1.7.7].Therefore, by [4, Theorem 1.1.6], X has an open cover B such that, foreach U ∈ B , dim( ∂U ) ≤ dim( X ) − . Now suppose { U , U , . . . , U m } is an open cover of X such that dim( ∂U i ) ≤ dim( X ) − ≤ i ≤ m , we define sets { V i : 1 ≤ i ≤ m } inductivelyby V := U , and V k := U k \ [ i
Let A be a continuous C ( X )-algebra such thateach fiber of A is rationally K -stable. By Theorem 2.5, we assumethat dim( X ) ≥
1, and we assume that A ( Y ) is rationally K -stable forany closed subset Y ⊂ X with dim( Y ) ≤ dim( X ) −
1. We now showthat the map ι ∗ ⊗ id : π j ( b U ( M n ( A )) ) ⊗ Q → π j ( b U ( M n +1 ( A )) ) ⊗ Q is an isomorphism for j ≥ n ≥
1. For simplicity of notation, weassume that n = 1.We first prove injectivity. Fix [ f ] ∈ π j ( b U ( A )) such that [ ι ◦ f ] has order n in π j ( b U ( M ( A ))), then we wish to prove that [ f ] has finite order in π j ( b U ( A )). For this consider F : [0 , → C ∗ ( S j , b U ( M ( A ))) such that F (0) = 0 , F (1) = µ M ( A ) n ( ι ◦ f ) . For x ∈ X , by Lemma 2.10, there is a closed neighbourhood Y x of x , N x ∈ N and a path L Y x : [0 , → C ∗ ( S j , b U ( A ( Y x ))) such that L Y x (0) = 0 , L Y x (1) = µ A ( Y x ) N x n ( π Y x ◦ f )and ι Y x ◦ L Y x ∼ h µ M ( A ( Y x )) N x ( η Y x ◦ F ) in C ∗ ( S j , b U ( M ( A ( Y x )))). We maychoose Y x to be the closure of a basic open set U x such that dim( ∂U x ) ≤ dim( X ) −
1. Since X is compact, we may choose a finite subcover { U , U , . . . , U m } . Now define { V , V , . . . , V m } and { W , W , . . . , W m − } as in Remark 2.11. We observe that each V i is a closed set such that µ A ( V i ) N i n ( π V i ◦ f ) ∼ C ∗ ( S j , b U ( A ( V i ))) since V i ⊂ U i for all 1 ≤ i ≤ m .Note that W = V ∩ V , and dim( W ) ≤ dim( X ) −
1. By induction hy-pothesis, A ( W ) is rationally K -stable. Let H i : [0 , → C ∗ ( S j , b U ( A ( V i ))), i = 1 , H i (0) = 0 , H i (1) = µ A ( V i ) N i n ( π V i ◦ f ), and ι V i ◦ H i ∼ h µ M ( A ( V i )) N i ( η V i ◦ F ) . Setting M := lcm( N , N ), we may assume that H i : [0 , → C ∗ ( S j , b U ( A ( V i ))), i = 1 , H i (0) = 0 , H i (1) = µ A ( V i ) Mn ( π V i ◦ f ), and ι V i ◦ H i ∼ h µ M ( A ( V i )) M ( η V i ◦ F ) . Let S : [0 , → C ∗ ( S j , b U ( A ( W ))) be the path S := ( π V W ◦ H ) • ( π V W ◦ H ) . Note that S (0) = S (1) = 0, so S is a loop in C ∗ ( S j , b U ( A ( W ))), and ι W ◦ S = ( η V W ◦ ι V ◦ H ) • ( η V W ◦ ι V ◦ H ) ∼ h µ M ( A ( W )) M ( η W ◦ F • ( η W ◦ F )) ∼ h . Also, since A ( W ) is rationally K -stable ι ∗ W ⊗ id : π ( C ∗ ( S j , b U ( A ( W )))) ⊗ Q → π ( C ∗ ( S j , b U ( M ( A ( W ))) ) ⊗ Q ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 19 is an isomorphism. Hence, there exists m ∈ N such that m [ S ] = 0 in π ( C ∗ ( S j , b U ( A ( W )))). Thus π V W ◦ µ A ( V ) m ( H ) = µ A ( W ) m ( π V W ◦ H ) ∼ h µ A ( W ) m ( π V W ◦ H ) = π V W ◦ µ A ( V ) m ( H )in C ∗ ( S j , b U ( A ( W ))). Now, by Lemma 1.5, and [19, Theorem 3.9], wehave a pullback diagram C ∗ ( S j , A ( V ∪ V )) π V ∪ V V / / π V ∪ V V (cid:15) (cid:15) C ∗ ( S j , A ( V )) π V W (cid:15) (cid:15) C ∗ ( S j , A ( V )) π V W / / C ∗ ( S j , A ( W )) . As mentioned before, this induces a pullback diagram of groups ofquasi-unitaries. Furthermore, the map π V W : b U ( A ( V )) → b U ( A ( W ))is a Serre fibration. Thus, by Lemma 2.7, µ A ( V ∪ V ) mMn ( π V ∪ V ◦ f ) ∼ h C ∗ ( S j , b U ( A ( V ∪ V ))). Thus, mM n [ π V ∪ V ◦ f ] = 0, so that [ π V ∪ V ◦ f ]has finite order in π j ( b U ( A ( V ∪ V ))).Now observe that W = ( V ∪ V ) ∩ V , and dim( W ) ≤ dim( X ) − V by V ∪ V and V by V in the above argument, we may re-peat the earlier procedure. By induction on the number of elements inthe finite subcover, we conclude that [ f ] has finite order in π j ( b U ( A )),as required.We now prove surjectivity of ι ∗ ⊗ id. Choose [ u ] ∈ π j ( b U ( M ( A ))) and m ∈ Z non-zero. We wish to construct an element [ ω ] ∈ π j ( b U ( A )) and q ∈ Q such that ι ∗ ⊗ id ([ ω ] ⊗ q ) = [ u ] ⊗ m . So, fix x ∈ X . Then by rationally K -stability of A ( x ) (as in the proofof Theorem 2.5), there is a closed neighbourhood Y x of x , a naturalnumber L x ∈ N , and a quasi-unitary c x ∈ C ∗ ( S j , b U ( A ( Y x )) such that µ M ( A ( Y x )) L x ( η Y x ◦ u ) ∼ h ι Y x ◦ c x . As in the first part of the proof, we may reduce to the case where X = V ∪ V , and there are quasi-unitaries c V ∈ C ∗ ( S j , b U ( A ( V ))) , c V ∈ C ∗ ( S j , b U ( A ( V ))) such that µ M ( A ( V i )) L i ( η V i ◦ u ) ∼ ι V i ◦ c V i in C ∗ ( S j , b U ( M ( A ( V i )))) i = 1 , and if W := V ∩ V , then dim( W ) ≤ dim( X ) −
1. Furthermore, byreplacing the { L i } by their least common multiple, we may assumethat L = L =: L . Now, fix paths H i : [0 , → C ∗ ( S j , b U ( M ( A ( V i ))))such that H (0) = ι V ◦ c V , H (1) = µ M ( A ( V )) L ( η V ◦ u ) H (0) = µ M ( A ( V )) L ( η V ◦ u ) , H (1) = ι V ◦ c V . Consider the path F : [0 , → C ∗ ( S j , b U ( M ( A ( W )))) given by F := ( η V W ◦ H ) • ( η V W ◦ H ) . Then F (0) = ι W ◦ π V W ◦ c V and F (1) = ι W ◦ π V W ◦ c V . Then since A ( W ) is rationally K -stable, by Proposition 2.9, there exists a path F ′ : [0 , → C ∗ ( S j , b U ( A ( W ))) and a natural number N ∈ N such that F ′ (0) = µ A ( W ) N ( π V W ◦ c V ) , F ′ (1) = µ A ( W ) N ( π V W ◦ c V )and ι W ◦ F ′ is path homotopic to µ M ( A ( W )) N ( F ) in C ∗ ( S j , b U ( M ( A ( W )))).The map π V W : C ∗ ( S j , b U ( A ( V ))) → π V W ( C ∗ ( S j , b U ( A ( V )))) is a fibra-tion, so there is a path F ′′ : [0 , → C ∗ ( S j , b U ( A ( V ))) such that F ′′ (1) = µ A ( V ) N ( c V ) , and π V W ◦ F ′′ = F ′ . Define e V := F ′′ (0) so that π V W ◦ e V = µ A ( W ) N ( π V W ◦ c V ) . Recall that, given a path G in a topological space, the path e G is definedby Eq. (6). Define H : [0 , → C ∗ ( S j , b U ( M ( V ))) as H := µ M ( A ( V )) N ( H ) • ^ ( ι V ◦ F ′′ ) . Then, H (0) = µ M ( A ( V )) NL ( η V ◦ u ) , H (1) = ι V ◦ e V , and η V W ◦ H = η V W ◦ ( µ M ( A ( V )) N ( H )) • ^ ( ι W ◦ F ′ ) . Also η V W : C ∗ ( S j , b U ( M ( A ( V )))) → η V W ( C ∗ ( S j , b U M (( A ( V )))) is afibration, thus η V W ◦ ( µ M ( A ( V )) N ( H )) has a lift, denoted by T : [0 , → C ∗ ( S j , b U ( M ( A ( V )))) so that T (0) = µ M ( A ( V )) NL ( η V ◦ u ) . Then, letting G := µ M A ( V N ( H ) • T gives η V W ◦ G = µ M ( A ( W )) N ( F ).Again by the above fibration map, since η V W ◦ G = µ M ( A ( W )) N ( F ) ∼ h ι W ◦ ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 21 F ′ , by calculation done in Lemma 2.7, ^ ι W ◦ F ′ has a lift in C ∗ ( S j , b U ( M ( V ))),denoted by T ′ . Then η V W ◦ ( T • T ′ ) = η V W ◦ ( µ M ( A ( V )) N ( H )) • ^ ( ι W ◦ F ′ ) . As before, C ∗ ( S j , A ) is a pullback C ∗ ( S j , A ) π V / / π V (cid:15) (cid:15) C ∗ ( S j , A ( V )) π V W (cid:15) (cid:15) C ∗ ( S j , A ( V )) π V W / / C ∗ ( S j , A ( W ))so that ω := ( µ A ( V ) N ( c V ) , e V ) defines a quasi-unitary in C ∗ ( S j , A ), and ι ◦ ω ∼ µ M ( A ) NL ( u ) in C ∗ ( S j , b U ( M ( A ))), where the path is given by thepair ( H , T • T ′ ). Hence, for q := 1 / ( mN L ), we have ι ∗ ⊗ id([ ω ] ⊗ q ) = [ u ] ⊗ m as required. (cid:3) We conclude this section with a discussion on the extent to which theconverse of Theorem A holds.
Proposition 2.12.
Let X be a locally compact, Hausdorff space, and A be a C ∗ -algebra. If A is rationally K -stable, then so is C ( X ) ⊗ A .The converse is true if X is a finite CW-complex.Proof. If A is rationally K -stable, we wish to show that C ( X ) ⊗ A isrationally K -stable. By appealing to the five lemma (as in [23, Lemma2.1]), we may assume that X is compact. Now, X is an inverse limit ofcompact metric spaces ( X i ) by [16], so that C ( X ) ⊗ A ∼ = lim C ( X i ) ⊗ A .Since the functors F j are continuous (Proposition 1.2), we may assumethat X itself is a compact metric space. Any metric space can, in turn,be written as an inverse limit of finite CW-complexes [7]. Therefore,we may further assume that X is a finite CW-complex. In that case,by [15, Theorem 4.20], one has(7) F j ( C ( X, A )) ∼ = M n ≥ j H n − j ( X ; F j ( A ))where the isomorphism is natural. Since the map ι ∗ : F j ( M n − ( A )) → F j ( M n ( A )) is an isomorphism, it follows that ι ∗ : F j ( C ( X, M n − ( A ))) → F j ( C ( X, M n ( A ))) is an isomorphism as well. Hence, C ( X ) ⊗ A is ra-tionally K -stable. Now suppose X is a finite CW-complex and C ( X ) ⊗ A is K -stable.Then, F j ( C ( X, M n − ( A ))) ∼ = F j ( C ( X, M n ( A )))and the isomorphism of Eq. (7) is component-wise. This implies that H n − j ( X ; F j ( M n − ( A ))) ∼ = H n − j ( X ; F j ( M n ( A ))) , ∀ n ≥ j For any connected H -space Y , as in Example 2.1, there is a fibrationsequence C ∗ ( X, Y ) → C ( X, Y ) → Y , which induces a short exactsequence of rational homotopy groups(8) 0 → F j ( C ∗ ( X, Y )) → F j ( C ( X, Y )) → F j ( Y ) → Y = b U ( M k ( A )) and apply [15, Theorem 4.20] to get F j ( C ∗ ( X, M k ( A ))) ∼ = M n ≥ j e H n − j ( X ; F j ( M k ( A )))and the isomorphism is natural. Hence, we conclude that F j ( C ∗ ( X, M n − ( A ))) ∼ = F j ( C ∗ ( X, M n ( A )))as well. By Eq. (8) and the five lemma, we conclude that A is rational K -stable. (cid:3) In [22, Theorem B], we proved that, for an AF-algebra, rational K -stability is equivalent to K -stability. Combining this fact with Proposition 2.12,and [23, Theorem A], we have Corollary 2.13.
Let X be a finite CW-complex, and A be an AF -algebra. Then, C ( X ) ⊗ A is K -stable if and only if A is K -stable. The next example shows that the converse of Theorem A need not holdfor arbitrary continuous C ( X )-algebras Example 2.14.
Let D := M ∞ denote the UHF algebra of type 2 ∞ ,and let D := D ⊕ M ( C ). Consider the C [0 , A := { ( f, g ) ∈ C [0 , / ⊗ D ⊕ C [1 / , ⊗ D : Φ( f ) = g (1 / } where Φ : C [0 , / ⊗ D → D is given by Φ( f ) = ( f (1 / , A is a continuous C [0 , A may be described as a pullback A / / (cid:15) (cid:15) C [ , ⊗ D (cid:15) (cid:15) C [0 , ] ⊗ D / / D ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 23 where ev is the evaluation at 1 /
2. The Mayer-Vietoris theorem [21,Theorem 4.5] for the functor F m gives a long exact sequence . . . → F m ( A ) → F m ( D ) ⊕ F m ( D ) ev ∗ − Φ ∗ −−−−→ F m ( D ) → . . . where ev ∗ : F m ( D ) → F m ( D ) is the identity map and Φ ∗ : F m ( D ) → F m ( D ) is given as Φ ∗ ( r ) = ( r, ∗ − Φ ∗ ) : F m ( D ) ⊕ F m ( D ) → F m ( D ) is given by(ev ∗ − Φ ∗ )(( a, b ) , c ) = ( a − c, b ) . Consider the case where m is odd: By [22, Lemma 3.2], F m − ( D i ) = F m +1 ( D i ) = 0 for i = 1 ,
2. Hence, the above long exact sequence boilsdown to0 → F m ( A ) → F m ( D ) ⊕ F m ( D ) ev ∗ − Φ ∗ −−−−→ F m ( D ) → . . . Thus, there is a natural isomorphism F m ( A ) = ker(ev ∗ − Φ ∗ ) ∼ = F m ( D )Similarly, F m ( M ( A )) ∼ = F m ( M ( D )) and the following diagram com-mutes F m ( A ) ∼ = / / ι A (cid:15) (cid:15) F m ( D ) ι D (cid:15) (cid:15) F m ( M ( A )) ∼ = / / F m ( M ( D )) . Since D is rationally K -stable by [22, Theorem B], it follows that ι A is an isomorphism. Doing the same for the inclusion map M n ( A ) ֒ → M n +1 ( A ), we conclude that the map F m ( M n ( A )) → F m ( M n +1 ( A )) isan isomorphism if m is odd.Now suppose m is even: The above long exact sequence reduces to F m − ( A ) → F m − ( D ) ⊕ F m − ( D ) ev ∗ − Φ ∗ −−−−→ F m − ( D ) → F m ( A ) → F m ( A ) ∼ = coker(ev ∗ − Φ ∗ ). Now, by [22, Theorem A], it followsthat F m − ( D ) ∼ = Q for all even m , and F m − ( D ) ∼ = ( Q ⊕ Q : m = 2 , Q : m > . Thus, elementary linear algebra proves that ev ∗ − Φ ∗ is surjective, sothat F m ( A ) = 0. Similarly, F m ( M n ( A )) = 0 for all n ≥ m is even). Thus, we conclude that A is rationally K -stable. However, one of itsfibers (namely D ) is not rationally K -stable because it has a non-zerofinite dimensional representation [22, Theorem B].3. An application to Crossed Product C*-algebras
As an application of our earlier results, we wish to show that the classof (rationally) K -stable C*-algebras is closed under the formation ofcertain crossed products. To begin with, we fix some conventions. Inwhat follows, G will denote a compact, second countable group, and A will denote a separable C*-algebra. By an action of G of A , we meana continuous group homomorphism α : G → Aut( A ), where Aut( A ) isequipped with the point-norm topology. We write σ : G → Aut( C ( G ))for the left action of G on C ( G ), given by σ s ( f )( t ) := f ( s − t ).The notion of Rokhlin dimension was invented by Hirshberg, Winterand Zacharias [12] for actions of finite groups (and the integers). Thedefinition for compact, second countable groups is due to Gardella [8].The ‘local’ definition we give below is different from the original, butis equivalent due to [8, Lemma 3.7] (See also [27, Lemma 1.5]). Definition 3.1.
Let G be a compact, second countable group, and let A be a separable C*-algebra. We say that an action α : G → Aut( A )has Rokhlin dimension d (with commuting towers) if d is the leastinteger such that, for any pair of finite sets F ⊂ A, K ⊂ C ( G ), andany ǫ >
0, there exist ( d + 1) contractive, completely positive maps ψ , ψ , . . . , ψ d : C ( G ) → A satisfying the following conditions:(1) For f , f ∈ K such that f ⊥ f , k ψ j ( f ) ψ j ( f ) k < ǫ for all0 ≤ j ≤ d .(2) For any a ∈ F and f ∈ K , k [ ψ j ( f ) , a ] k < ǫ for all 0 ≤ j ≤ d .(3) For any f ∈ K and s ∈ G, k α s ( ψ j ( f )) − ψ j ( σ s ( f )) k < ǫ for all0 ≤ j ≤ d .(4) For any a ∈ F, k P dj =0 ψ j (1 C ( G ) ) a − a k < ǫ .(5) For any f , f ∈ K , k [ ψ j ( f ) , ψ k ( f )] k < ǫ for all 0 ≤ j, k ≤ d .We denote the Rokhlin dimension (with commuting towers) of α bydim cRok ( α ). If no such integer exists, we say that α has infinite Rokhlindimension (with commuting towers), and write dim cRok ( α ) = + ∞ .We now describe the local approximation theorem due to Gardella,Hirshberg and Santiago [9] that will help prove the permanence resultwe are interested in. ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 25 Proposition 3.2. [9, Corollary 4.9]
Let G be a compact, second count-able group, X be a compact Hausdorff space and A be a separable C*-algebra. Let G y X be a continuous, free action of G on X , and α : G → Aut ( A ) be an action of G on A . Equip the C*-algebra C ( X, A ) with the diagonal action of G , denoted by γ . Then, the crossed prod-uct C*-algebra C ( X, A ) ⋊ γ G is a continuous C ( X/G ) -algebra, each ofwhose fibers are isomorphic to A ⊗ K ( L ( G )) . In the context of Proposition 3.2, the natural inclusion map ρ : A → C ( X, A ) is a G -equivariant ∗ -homomorphism. Hence, it induces a map ρ : A ⋊ α G → C ( X, A ) ⋊ γ G . To describe the nature of this map,we need the next definition, which is due to Barlak and Szabo [1].Once again, we choose to work with the local definition as it is moreconvenient for our purpose. Definition 3.3.
Let A and B be separable C*-algebras. A ∗ -homomorphism ϕ : A → B is said to be sequentially split if, for every compactset F ⊂ A , and for every ǫ >
0, there exists a ∗ -homomorphism ψ = ψ F,ǫ : B → A such that k ψ ◦ φ ( a ) − a k < ǫ for all a ∈ F .The next theorem, due to Gardella, Hirshberg and Santiago [9, Propo-sition 4.11] is an important structure theorem that allows one to provepermanence results concerning crossed products with finite Rokhlindimension (with commuting towers). Theorem 3.4.
Let α : G → Aut ( A ) be an action of a compact, secondcountable group on a separable C*-algebra such that dim cRok ( α ) < ∞ .Then, there is exists a compact metric space X and a free action G y X such that the canonical embedding ρ : A ⋊ α G → C ( X, A ) ⋊ γ G is sequentially split. Furthermore, if G finite dimensional, then X maybe chosen to be finite dimensional as well. In light of Theorem 3.4, we now show that the property of being ra-tionally K -stable ( K -stable) passes from the target algebra B to the do-main algebra A , in the presence of a sequentially split ∗ -homomorphism.To this end, we fix the following notations, given ∗ -homomorphism ϕ : A → B , ϕ n : M n ( A ) → M n ( B ) represents the inflation of ϕ , givenby ϕ n (( a i,j )) = ( ϕ ( a i,j )). Furthermore ι B : B → M ( B ) represents thecanonical inclusion. Proposition 3.5.
Let A and B be separable C*-algebras, and ϕ : A → B be a sequentially split ∗ -homomorphism. If B is rationally K -stable( K -stable), then so is A .Proof. Since the proof of both cases is entirely similar, we only provethat rational K -stability passes from B to A . As before, we need toshow that the map( ι A ) ∗ ⊗ id : π j ( b U ( M n ( A ))) ⊗ Q → π j ( b U ( M n +1 ( A ))) ⊗ Q is an isomorphism for all j ≥
1, and n ≥
1. If ϕ : A → B is sequentiallysplit, then so is ϕ n : M n ( A ) → M n ( B ), so we may assume without lossof generality that n = 1.We first show that ( ι A ) ∗ ⊗ id is injective. So suppose [ f ] ⊗ q ∈ π j ( b U ( A )) ⊗ Q is such that [ ι A ◦ f ] ⊗ q = 0 in π j ( b U ( M ( A ))). Then, [ ι A ◦ f ] has finiteorder in π j ( b U ( M ( A ))), which implies [ ϕ ◦ ι A ◦ f ] = [ ι B ◦ ϕ ◦ f ] hasfinite order in π j ( b U ( M ( B )) ). Since B is rationally K -stable, [ ϕ ◦ f ]also has finite order in π j ( b U ( B )). Let F := { f ( x ) : x ∈ S j } , which is acompact set in A , so there exists a ∗ -homomorphism ψ = ψ F, : B → A such that k ψ ◦ ϕ ( a ) − a k < a ∈ F . Hence, k ψ ◦ ϕ ◦ f − f k < b U ( C ∗ ( S j , A )). Thus, by Lemma 2.3, we conclude that[ ψ ◦ ϕ ◦ f ] = [ f ]in π j ( b U ( A )). However, since [ ϕ ◦ f ] has finite order in π j ( b U ( B )),[ ψ ◦ ϕ ◦ f ] = [ f ] has finite order in π j ( b U ( A )). Hence, [ f ] ⊗ q = 0,proving that ( ι A ) ∗ ⊗ id is injective.For surjectivity, fix an element [ u ] ∈ π j ( b U ( M ( A )) and m ∈ Z . Wewish to construct elements [ ω ] ∈ π j ( b U ( A )) and q ∈ Q such that(( ι A ) ∗ ⊗ id) ([ ω ] ⊗ q ) = [ u ] ⊗ m . Now, [ ϕ ◦ u ] ⊗ m ∈ π j ( b U ( M ( B ))) ⊗ Q . Since B is rationally K -stable,there exists [ g ] ∈ π j ( b U ( B )) and n ∈ Z such that(( ι B ) ∗ ⊗ id) (cid:18) [ g ] ⊗ n (cid:19) = [ ϕ ◦ u ] ⊗ m . Again, as in previous calculations, there exists N , N ∈ N such that(9) N [ ι B ◦ g ] = N [ ϕ ◦ u ] ATIONAL K-STABILITY OF CONTINUOUS C ( X )-ALGEBRAS 27 in π j ( b U ( M ( B ))). Now, fix F := { u ( x ) : x ∈ S j } , so we get a ∗ -homomorphism ψ F : M ( B ) → M ( A ) such that k ψ F ◦ ϕ ◦ u ( x ) − u ( x ) k < for all x ∈ S j . Hence,(10) [ ψ F ◦ φ ◦ u ] = [ u ]in π j ( b U ( M ( A ))). Now, we write u = ( u i,j ) ≤ i,j ≤ , and take K = { u i,j ( x ) : 1 ≤ i, j ≤ , x ∈ S j } ⊂ A . Then K is compact, so we get a ∗ -homomorphism ψ K : B → A such that k ψ K ◦ ϕ ◦ u i,j ( x ) − u i,j ( x ) k < x ∈ S j and 1 ≤ i, j ≤
2. Thus k ( ψ K ) ◦ ϕ ◦ u ( x ) − u ( x ) k < for all x ∈ S j . Therefore, k ψ F ◦ ϕ ◦ u ( x ) − ( ψ K ) ◦ φ ◦ u ( x ) k < x ∈ S j , so that [ ψ F ◦ ϕ ◦ u ] = [( ψ K ) ◦ ϕ ◦ u ] in π j ( b U ( M ( A ))).Now, from Eq. (9) and Eq. (10), N [( ψ K ) ◦ ι B ◦ g ] = N [( ψ K ) ◦ φ ◦ u ] = N [ ψ F ◦ φ ◦ u ] = N [ u ] . Since ( ψ K ) ◦ ι B ◦ g = ι A ◦ ψ K ◦ g , we have N [ ι A ◦ ψ K ◦ g ] = N [ u ] . Therefore, if ω := ψ K ◦ g and q := N N m , then( ι A ) ∗ ⊗ id ([ ω ] ⊗ q ) = [ u ] ⊗ m proving that ( ι A ) ∗ ⊗ id is surjective. (cid:3) We are now in a position to complete the proof of Theorem B.
Corollary 3.6.
Let α : G → Aut ( A ) be an action of a compact Liegroup on a separable C*-algebra A such that dim cRok ( α ) < ∞ . If A isrationally K -stable ( K -stable), then so is A ⋊ α G .Proof. We first discuss the case of K -stability: Let X be the (finite di-mensional) metric space space obtained from Theorem 3.4. By Proposition 3.2, C ( X, A ) ⋊ γ G is a continuous C ( X/G )-algebra, each of whose fibers areisomorphic to A ⊗ K ( L ( G )), and are hence K -stable. Since X is com-pact and metrizable, so is X/G . Furthermore, since G is a compactLie group, it follows thatdim( X/G ) ≤ dim( X ) < ∞ by [18, Corollary 1.7.32]. By [23, Theorem A], we conclude that C ( X, A ) ⋊ γ G is K -stable, and hence A ⋊ α G is K -stable by Proposition 3.5.The argument for rational K -stability is entirely similar, except thatwe apply Theorem A instead of [23, Theorem A]. (cid:3) Acknowledgements
The first named author is supported by UGC Junior Research Fellow-ship No. 1229, and the second named author was partially supportedby the SERB (Grant No. MTR/2020/000385).
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