A topological invariant for continuous fields of Cuntz algebras II
aa r X i v : . [ m a t h . OA ] O c t A topological invariant for continuous fields of Cuntzalgebras II
Taro Sogabe
Graduate School of Science, Kyoto University, [email protected]
October 5, 2020
Abstract
We investigate an invariant for continuous fields of the Cuntz algebra O n +1 introducedin [18], and find a way to obtain a continuous field of M n ( O ∞ ) from that of O n +1 usingthe construction of the invariant. By Brown’s representability theorem, this gives a bi-jection from the set of the isomorphism classes of continuous fields of O n +1 to those of M n ( O ∞ ). As a consequence, we obtain a new proof for M. Dadarlat’s classification resultof continuous fields of O n +1 arising from vector bundles, which corresponds to those of M n ( O ∞ ) stably isomorphic to the trivial field. Our purpose is to investigate the invariant for continuous fields of the Cuntz algebrasintroduced in [18]. The Cuntz algebra O n +1 is a typical example of a Kirchberg algebra,and its continuous fields over a finite CW-complex X are classified in [5] when the coho-mology groups H ∗ ( X, Z ) do not admit n − torsion. All continuous fields classified in [5]are constructed via the Cuntz–Pimsner algebras, and it is proved in [5] that, for general X , not every continuous field of O n +1 is given by the Cuntz–Pimsner construction.Recently, M. Dadarlat and U. Pennig introduced a generalized cohomology E ∗ D in [9]for every strongly self-absorbing C*-algebra D satisfying the UCT including the infiniteCuntz algebra O ∞ . In [18], using the reduced cohomology ¯ E ∗O ∞ , we define an invariant b O ∞ of continuous fields of O n +1 , and show that a continuous field O over a finite CW-complex X is given via the Cuntz–Pimsner algebra if and only if b O ∞ ([ O ]) = 0 ∈ ¯ E O ∞ ( X ).By [6], the set of the isomorphism classes of continuous fields of O n +1 over a finite CW-complex X is identified with the homotopy set [ X, BAut( O n +1 )], and the invariant is amap b O ∞ : [ X, BAut( O n +1 )] → ¯ E O ∞ ( X ).In this paper, we construct a natural transformation T X : [ X, BAut( O n +1 )] → [ X, BAut( M n ( O ∞ ))]which turns out to be bijective by Brown’s representability theorem (Theorem 3.8). Sincethe homotopy set [ X, BAut( M n ( O ∞ ))] is identified with the set of the isomorphism classesof locally trivial continuous fields of M n ( O ∞ ), the map B( η ) ∗ : [ X, BAut( M n ( O ∞ ))] ∋ [ B ] [ K ⊗ B ] ∈ ¯ E O ∞ ( X ) is defined (see Section 2.2), and we have − b O ∞ = B( η ) ∗ ◦ T X . ince the inverse image b − O ∞ (0) is equal to the set of the Cuntz–Pimsner algebras ofvector bundles classified in [5] (see [18, Sec. 4]), M. Dadarlat’s classification result [5, Th.5.3], which enables us to count the cardinality of the set of the isomorphism classes ofthose Cuntz–Pimsner algebras, can be proved by counting the cardinality of the set of theisomorphism classes of the continuous fields of M n ( O ∞ ) stably isomorphic to the trivialfield (see Remark 2.4, Corollary 3.10).We also investigate the map T SX for the reduced suspension SX , and show that themap gives a group isomorphism T SX : [ X, Aut( O n +1 )] → [ X, Aut( M n ( O ∞ ))] (Corollary3.9). Acknowledgements
The author would like to thank Prof. M. Izumi for suggesting to investigate the K ( X )-module homomorphism in Remark 2.7 and for his support and encouragement. Theauthor also would like to thank Prof. U. Pennig for helpful discussions about Lemma 2.2,2.6, Remark 2.7 and for many stimulating conversations. Let K be the C*-algebra of compact operators on the separable infinite dimensional Hilbertspace, and let M n be the n by n matrix algebra. For a C*-algebra A , we denote by K i ( A )the i-th K-group and denote by [ p ] ∈ K ( A ) (resp. [ u ] ∈ K ( A )) the class of theprojection p (resp. the unitary u ). If A is unital, we denote by 1 A the unit and by U ( A )the group of unitary elements. Let C ( X ) be the C*-algebra of all continuous functionson X . We write K i ( X ) = K i ( C ( X )) , ˜ K i ( X ) = K i ( C ( X, x )) where C ( X, x ) is the setof functions vanishing at x ∈ X . We refer to [1] for the K-groups.Let ( X, x ) and ( Y, y ) be two pointed finite CW-complexes, and let [ X, Y ] (resp.[
X, Y ] ) be the set of the homotopy classes of the continuous maps (resp. the set of thebase point preserving homotopy classes of the base point preserving continuous maps).The i-th homotopy group of X is π i ( X ) := [ S i , X ] , and, for Y with π ( Y ) = π ( Y ) = 0,the natural map [ X, Y ] → [ X, Y ] is bijective (see [11, Th. 6.57]).We refer to [17, 6] for the definition of the continuous C ( X )-algebras. Let P → X be a principal Aut( A ) bundle, and let A be the associated bundle P ×
Aut( A ) A . Then,the section algebra Γ( X, A ) is a locally trivial continuous C ( X )-algebra. We identifythe section algebra with the associated bundle, and write A by abuse of notation. Let A x denote the image of the evaluation map ev x : A → A /C ( X, x ) A ∼ = A . We alwaysassume that the fiber A x is nuclear. Since those principal Aut( A ) bundles are classifiedby [ X, BAut( A )], we denote the C ( X )-linear isomorphism class of the C ( X )-algebra by[ A ] ∈ [ X, BAut( A )]. For two locally trivial continuous C ( X )-algebras A , B , we can definethe tensor product A ⊗ C ( X ) B (see [2]).Let E n +1 be the universal C*-algebra, called the Cuntz–Toeplitz algebra, generatedby n + 1 isometries with mutually orthogonal ranges. It is known that the unital map C → E n +1 is a KK-equivalence. Let { T i } n +1 i =1 be the canonical generators, and let e :=1 − P n +1 i =1 T i T ∗ i be the minimal projection which generates the only non-trivial ideal of E n +1 isomorphic to K . The quotient algebra O n +1 := E n +1 / K is called the Cuntz algebra. Wedenote by O ∞ the universal C*-algebra generated by countably infinite isometries withmutually orthogonal ranges. The inclusion K → E n +1 gives the map K ( K ) = Z − n −−→ ( E n +1 ) = Z , and one has K ( O n +1 ) = Z n , K ( O ∞ ) = Z , K ( O n +1 ) = K ( O ∞ ) = 0(see [4]). One has O n +1 ∼ = O n +1 ⊗ O ∞ and O n +1 ∼ = ( E n +1 ⊗ O ∞ ) / ( K ⊗ O ∞ ) by theclassification result of the Kirchberg algebras.The homotopy groups of Aut( O n +1 ) and Aut( M n ( O ∞ )) are given by [8, Th. 5.9] : π k (Aut( O n +1 )) = π k (Aut( M n ( O ∞ ))) = 0 , π k +1 (Aut( O n +1 )) = π k +1 (Aut( M n ( O ∞ ))) = Z n ,k ≥
0. Thus, two sets [ S k , BAut( O n +1 )] , [ S k , BAut( M n ( O ∞ ))] have the same cardinality,and one has [ X, BAut( O n +1 )] = [ X, BAut( O n +1 )], [ X, BAut( M n ( O ∞ ))] = [ X, BAut( M n ( O ∞ ))]. We briefly explain the cohomology group ¯ E O ∞ ( X ). Recall that O ∞ is a strongly self-absorbing C*-algebra, in other words, there is a continuous path of unitary { u t } t ∈ [0 , ⊂O ⊗ ∞ and an isomorphism φ : O ∞ → O ⊗ ∞ satisfying lim t → || φ ( d ) − u t (1 ⊗ d ) u ∗ t || = 0. Werefer to [19] for the properties of the strongly self-absorbing C*-algebras. The followingmap ∆ X : ( C ( X ) ⊗ O ∞ ) ⊗ ∋ f ( x ) ⊗ f ( y ) φ − ( f ( x ) ⊗ f ( x )) ∈ C ( X ) ⊗ O ∞ gives K ( C ( X ) ⊗ O ∞ ) a ring structure which coincides with the ring structure of K ( X )coming from the tensor products of vector bundles. Let K ( X ) × := ± K ( X ) denotethe group of the invertible elements. Theorem 2.1 ([9, Th. 2.22, 3.8, Lem. 2.8, Cor. 3.9]) . Let X be a connected compactmetrizable space, and let Aut ( K ⊗O ∞ ) be the path component of Aut( K ⊗O ∞ ) containing id K ⊗O ∞ .1) For two continuous maps α, β : X → Aut( K ⊗ O ∞ ) which are identified with the C ( X ) -linear isomorphisms of C ( X ) ⊗ K ⊗ O ∞ , one has K (∆ X ) ◦ K ( α ⊗ β )([(1 C ( X ) ⊗ e ⊗ O ∞ ) ⊗ ] ) = K ( α ◦ β )([1 C ( X ) ⊗ e ⊗ O ∞ ] ) and the following injective map, whose range is K ( X ) × , is multiplicative :[ X, Aut( K ⊗ O ∞ )] ∋ [ α ] [ α (1 C ( X ) ⊗ e ⊗ O ∞ )] ∈ K ( C ( X ) ⊗ K ⊗ O ∞ ) .
2) The subgroup [ X, Aut ( K ⊗ O ∞ )] ⊂ [ X, Aut( K ⊗ O ∞ )] is identified with K ( X ) .3) The homotopy set E O ∞ ( X ) := [ X, BAut(
K ⊗ O ∞ )] has a group structure definedby the tensor product ⊗ C ( X ) of locally trivial continuous C ( X ) -algebras of K ⊗ O ∞ , and ¯ E O ∞ ( X ) := [ X, BAut ( K ⊗ O ∞ )] is a subgroup of E O ∞ ( X ) . For a continuous field [ A ] ∈ ¯ E O ∞ ( X ), one has another field denoted by A − satisfying − [ A ] = [ A − ] ∈ ¯ E O ∞ ( X ) (i.e., a C ( X )-linear isomorphism A⊗ C ( X ) A − → C ( X ) ⊗ K ⊗O ∞ exists). For a locally trivial continuous field B of M n ( O ∞ ), one has a locally trivialcontinuous field K ⊗ B of K ⊗ O ∞ . Since Aut( M n ( O ∞ )) is path connected, the grouphomomorphism η : Aut( M n ( O ∞ )) ∋ σ id K ⊗ σ ∈ Aut ( K ⊗ M n ( O ∞ ))gives a natural map B( η ) ∗ : [ X, BAut( M n ( O ∞ ))] ∋ [ B ] [ K ⊗ B ] ∈ ¯ E O ∞ ( X ). et Aut X ( A ) be the group of all C ( X )-linear isomorphisms of a locally trivial con-tinuous C ( X )-algebra A of K ⊗ O ∞ . We fix an isomorphism A ⊗ ( K ⊗ O ∞ ) ∋ f ⊗ d f ⊗ (1 C ( X ) ⊗ d ) ∈ A ⊗ C ( X ) ( C ( X ) ⊗ K ⊗ O ∞ ). This gives K ( A ) a K ( C ( X ) ⊗ K ⊗ O ∞ )-module structure by · [ q ] : K ( A ) ∋ [ p ] [ p ⊗ q ] ∈ K ( A⊗ C ( X ) ( C ( X ) ⊗ K ⊗O ∞ )) , [ q ] ∈ K ( C ( X ) ⊗ K ⊗O ∞ ) . Lemma 2.2.
In the above setting, the followings hold :
1) For every α ∈ Aut X ( A ) , there is an element a ∈ K ( X ) × with K ( α ) = · a ,2) For every a ∈ K ( X ) × , there is an element α ∈ Aut X ( A ) with · a = K ( α ) .Proof. We prove only 1). Fix an isomorphism θ : ( A − ⊗ C ( X ) A ) ⊗ C ( X ) → C ( X ) ⊗ K ⊗O ∞ .One has the following commutative diagram A ⊗ ( K ⊗ O ∞ ) (cid:15) (cid:15) α ⊗ id / / A ⊗ ( K ⊗ O ∞ ) (cid:15) (cid:15) A ⊗ C ( X ) ( C ( X ) ⊗ K ⊗ O ∞ ) α ⊗ id / / A ⊗ C ( X ) ( C ( X ) ⊗ K ⊗ O ∞ ) A ⊗ C ( X ) ( A − ⊗ A ) ⊗ ⊗ θ O O α ⊗ (id A− ⊗ id A ) ⊗ / / A ⊗ C ( X ) ( A − ⊗ A ) ⊗ . id ⊗ θ O O Since the flip automorphism σ : ( K ⊗ O ∞ ) ⊗ ∋ x ⊗ y y ⊗ x ∈ ( K ⊗ O ∞ ) ⊗ fixing theminimal projection ( e ⊗ O ∞ ) ⊗ is contained in Aut (( K ⊗ O ∞ ) ⊗ ) by Theorem 2.1, 2),one has K ( α ⊗ (id A − ⊗ id A ) ⊗ ) = K (id A ⊗ (id A − ⊗ α ⊗ id A − ⊗ id A )). We have anelement a := [ θ ◦ (id A − ⊗ α ⊗ id A − ⊗ id A ) ◦ θ − (1 C ( X ) ⊗ e ⊗ O ∞ )] ∈ K ( X ) × satisfying K ( α ) = · a .Using Theorem 2.1, similar argument proves the statement 2). Proposition 2.3.
Let ( X, x ) be a pointed, path connected, finite CW-complex. For [ A ] ∈ Im(B( η ) ∗ ) ⊂ ¯ E O ∞ ( X ) , we fix an isomorphism ϕ : A x ∼ = K ⊗ O ∞ . Then, thefollowing map is a well-defined bijection B( η ) − ∗ ([ A ]) ∋ [ B ] [[ ρ B ( e ⊗ B )] ] ∈ { [ p ] ∈ K ( A ) | K ( ϕ ◦ ev x )([ p ] ) = n } / ∼ , where ρ B : K ⊗ B → A is an isomorphism satisfying [( ϕ ◦ ev x ◦ ρ B )( e ⊗ B )] = n ∈ K ( K ⊗ O ∞ ) . Here, the equivalence relation is defined by [ p ] ∼ [ r ] ⇔ [ p ] · a = [ r ] forsome a ∈ K ( X ) .Proof. First, we show taht the map is well-defined. For two continuous fields B , B with[ B ] = [ B ] ∈ B( η ) − ∗ ([ A ]) and two C ( X )-linear isomorphisms ρ i : K ⊗ B i → A satisfying[( ϕ ◦ ev x ◦ ρ i )( e ⊗ B i )] = n , we show [ ρ ( e ⊗ B )] ∼ [ ρ ( e ⊗ B )] . Since we have anisomorphism γ : B → B , the following map β := ρ ◦ (id K ⊗ γ ) ◦ ρ − ∈ Aut X ( A )satisfies β ◦ ρ ( e ⊗ B ) = ρ ( e ⊗ B ), and Lemma 2.2 shows [ ρ ( e ⊗ B )] ∼ [ ρ ( e ⊗ B )] .By [7, Th. 1.1, 2.7], the element [ p ( A ) p ] is sent to [[ p ] ] and this map is surjective.Finally, we prove the injectivity. Suppose [ ρ ( e ⊗ B )] ∼ [ ρ ( e ⊗ B )] . Then, Lemma2.2 shows there is a map α ∈ Aut X ( A ) with [ α ( ρ ( e ⊗ B ))] = [ ρ ( e ⊗ B )] . Now thecancellation of the properly infinite full projections shows B ∼ = B . Remark 2.4.
For [ A ] = 0 , Proposition 2.3 allows us to count the number of the isomor-phism classes of the continuous fields of M n ( O ∞ ) stably isomorphic to C ( X ) ⊗ K ⊗ O ∞ ,which is equal to | ( n + ˜ K ( X )) / (1 + ˜ K ( X )) | . .3 The invariant b O ∞ An automorphism of E n +1 ⊗ O ∞ induces an automorphism of O n +1 ∼ = ( E n +1 ⊗ O ∞ ) / ( K ⊗O ∞ ), and one has a group homomorphism q : Aut( E n +1 ⊗ O ∞ ) → Aut( O n +1 ). Theorem 2.5 ([18, Cor. 3.15, Def. 4.1]) . Let X be a finite CW-complex.1) The group homomorphism q : Aut( E n +1 ⊗ O ∞ ) → Aut( O n +1 ) is a weak homotopyequivalence.2) For every continuous field O of O n +1 , one has an exact sequence of C ( X ) -algebras → A → E → O → , where we denote by E (resp. A ) a continuous field of E n +1 ⊗ O ∞ (resp. K ⊗ O ∞ ), andthe following map is well-defined : b O ∞ : [ X, BAut( O n +1 )] ∋ [ O ] [ A ] ∈ ¯ E O ∞ ( X ) . One has a bijection (B( q ) ∗ ) − : [ X, BAut( O n +1 )] ∋ [ O ] [ E ] ∈ [ X, BAut( E n +1 ⊗O ∞ )]by Theorem 2.5, 1), and the group homomorphism Aut( E n +1 ⊗ O ∞ ) ∋ α α | K ⊗O ∞ ∈ Aut ( K ⊗ O ∞ ) induces the map [ X, BAut( E n +1 ⊗ O ∞ )] ∋ [ E ] [ A ] ∈ ¯ E O ∞ ( X ) (see [18,Lem. 3.7]).Since E n +1 ⊗ O ∞ is KK-equivalent to C , the map id A − ⊗ A − ∋ f f ⊗ E ∈A − ⊗ C ( X ) E gives the isomorphism K (id A − ⊗
1) of their K -groups by [7, Th. 1.1]. Lemma 2.6.
Let ( X, x ) be a pointed, path connected, finite CW-complex. Let A and E be as in Theorem 2.5, 2), and let ι : A → E be the inclusion map. We fix isomorphisms ϕ : E x ∼ = E n +1 ⊗ O ∞ and π : ( A − ) x ∼ = K ⊗ O ∞ . Then, there exists a properly infinitefull projection p ∈ A − satisfying K ( π ◦ ev x )([ p ] ) = n ∈ K ( K ⊗ O ∞ ) and [1 p ( A − ) p ⊗ C ( X ) E ] ∈ Im( K ( p ( A − ) p ⊗ C ( X ) A ) K (id ⊗ ι ) −−−−−→ K ( p ( A − ) p ⊗ C ( X ) E )) . Proof.
Fix a C ( X )-linear isomorphism ψ : C ( X ) ⊗ K ⊗ O ∞ → A − ⊗ C ( X ) A with K (( π ⊗ ϕ | A x ) ◦ ev x )([ ψ (1 C ( X ) ⊗ e ⊗ O ∞ )] ) = 1 ∈ K (( K ⊗ O ∞ ) ⊗ ) . By [18, Th. 4.2] and [10, Th. 2.11], the algebra A − is isomorphic to a tensor product of aunital O ∞ -stable algebra and K . Therefore, one can find a properly infinite full projection p ∈ A − such that − [ p ] ∈ K ( A − ) is sent to K (id A − ⊗ ι )([ ψ (1 C ( X ) ⊗ e ⊗ O ∞ )] ) ∈ K ( A − ⊗ C ( X ) E ) by the isomorphism K (id A − ⊗ p ( A − ) p ⊗ C ( X ) A ֒ → A − ⊗ C ( X ) A and p ( A − ) p ⊗ C ( X ) E ֒ → A − ⊗ C ( X ) E giveisomorphisms of K-groups, and the statement is now proved. Remark 2.7.
Since K ( E ) = K ( X ) , the image of the K ( X ) -module homomorphism K ( A ) K ( ι ) −−−→ K ( E ) is an ideal of K ( X ) . In the case of Tor( H ∗ ( X ) , Z n ) = 0 , the ideal isthe complete invariant of the continuous field E / A (see [5, Sec. 2] and [18, Sec. 4]). Theelement − [ p ] ∈ K ( A − ) corresponds to KK X ( ι ) ∈ KK X ( A , E ) ∼ = K ( A − ) , and onecan identify K ( ι ) with the map K ( A ) ∋ [ r ]
7→ − [ r ⊗ p ] ∈ K ( A ⊗ C ( X ) A − ) ∼ = K ( X ) . By Lemma 2.6, we constructs an element [ p ( A − ) p ] ∈ [ X, BAut( M n ( O ∞ ))] from[ O ] ∈ [ X, BAut( O n +1 )]. In Section 3, we verify that this gives a natural transformationbetween two functors [ · , BAut( O n +1 )] and [ · , BAut( M n ( O ∞ ))] defined on the category ofthe pointed connected finite CW-complexes. .4 The homotopy sets [ X, Aut( O n +1 )] and [ X, Aut( M n ( O ∞ ))] We briefly recall [8, Th. 5.9]. Let B be a unital Kirchberg algebra with path connectedAut( B ), and let C ν := { f ∈ C [0 , ⊗ B | f (0) = 0 , f (1) ∈ C B } be the mapping cone of theunital map ν : C → B with the inclusion map j : SB := C (0 , ⊗ B → C ν B . Let α, l :( X, x ) → (Aut( B ) , id B ) be the base point preserving continuous maps where l : x id B is the constant map. These two maps define an element h α, l i ∈ KK ( C ν B, SC ( X, x ) ⊗ B )(see [8, p 123]), and the map [ X, Aut( B )] ∋ [ α ]
7→ h α, l i ∈ KK ( C ν , SC ( X, x ) ⊗ B ) isbijective.In the case of B = M n ( O ∞ ), one has KK ( C ν B, SC ( X, x ) ⊗ B ) = KK ( C ν B, SC ( X ) ⊗ B ), and the map j ∗ : KK ( C ν B, SC ( X ) ⊗ B ) → KK ( SB, SC ( X ) ⊗ B ) maps h α, l i to KK (id C (0 , ⊗ α ) − KK (id C (0 , ⊗ l ) = S ( η ∗ ([ α ]) − S : K ( X ) = KK ( B, C ( X ) ⊗ B ) → KK ( SB, SC ( X ) ⊗ B ) is the suspension isomorphism (see Theorem2.1 for the definition of η ∗ ). For another map β : ( X, x ) → (Aut( B ) , id B ), one has h α ◦ β, l i = h α ◦ β, α i + h α, l i = (id C ( R ) ⊗ α ) ∗ ( h β, l i ) + h α, l i = h β, l i ⊗ ( KK (id C (0 , ⊗ α ) −
1) + h α, l i + h β, l i = h α, l i + h β, l i + h β, l i · ( S − ◦ j ∗ )( h α, l i ) . The product h β, l i · ( S − ◦ j ∗ )( h α, l i ) is defined by KK ( C ν B, SC ( X ) ⊗ B ) × KK ( B, C ( X ) ⊗ B ) → KK ( C ν B, SC ( X × X ) ⊗ B ) ∆ X ∗ −−−→ KK ( C ν B, SC ( X ) ⊗ B ) . Theorem 2.8 ([8, Th. 6.3, Th. 5.9]) . We write
Ad : U ( C ( X ) ⊗ M n ( O ∞ )) ∋ v Ad v ∈ Map( X, Aut( M n ( O ∞ ))) .1) There is a short exact sequence of groups :0 → K ( C ( X ) ⊗ M n ( O ∞ )) ⊗ Z n Ad −−→ [ X, Aut( M n ( O ∞ ))] η ∗ −→ (1 + Tor( K ( X ) , Z n )) × → .
2) The multiplication a ⋆ b := a + b + b · ( S − ◦ j ∗ )( a ) makes ( KK ( C ν M n ( O ∞ ) , SC ( X ) ⊗ M n ( O ∞ )) , ⋆ ) a group with the following isomorphism [ X, Aut( M n ( O ∞ ))] ∋ [ β ]
7→ h β, l i ∈ KK ( C ν M n ( O ∞ ) , SC ( X ) ⊗ M n ( O ∞ )) . M. Izumi obtained similar results for [ X, Aut( O n +1 )]. Let δ : K ( C ( X ) ⊗ O n +1 ) → K ( X ) be the index map coming from the exact sequence C ( X ) ⊗ K → C ( X ) ⊗ E n +1 → C ( X ) ⊗ O n +1 . The map u : Aut( O n +1 ) → U ( O n +1 ) defined by u ( α ) := P n +1 i =1 α ( S i ) S ∗ i is aweak homotopy equivalence (see [6]), where the isometries S i are the canonical generatorsof O n +1 . Theorem 2.9 ([15, Th. 3.1]) . We define a multiplication of K ( C ( X ) ⊗ O n +1 ) by a ⋄ b := a + b − a · δ ( b ) .1) The map u ∗ : [ X, Aut( O n +1 )] → ( K ( C ( X ) ⊗ O n +1 ) , ⋄ ) is a group isomorphism.2) There is a short exact sequence of groups → K ( X ) ⊗ Z n → [ X, Aut( O n +1 )] − δ −−→ (1 + Tor( K ( X ) , Z n )) × → where the map − δ is defined by (1 − δ )([ α ]) := 1 − δ ([ u ( α )] ) . et Σ X be the unreduced suspension of X . Then, one has a bijection [Σ X, B G ] → [ X, G ] for every path connected group G (see [12, Cor. 8.3]), where an element [ α ] ∈ [ X, G ]is sent to the principal G -bundle whose clutching function is α : X → G . We denote byΓ α (Σ X ) A := { ( F , F ) ∈ ( C ([0 , × X ) ⊗ A ) ⊕ | F i (0) ∈ C ( X ) ⊗ A, F (1) = α ( F (1)) ∈ C ( X ) ⊗ A } the C (Σ X )-algebra whose isomorphism class in [Σ X, BAut( A )] corresponds to the element[ α ] ∈ [ X, Aut( A )]. We use another C*-algebra defined by M α ( X ) A := { f ∈ C ([0 , × X ) ⊗ A | f (0) ∈ C ( X ) ⊗ A, f (1) = α ( f (0)) ∈ C ( X ) ⊗ A } with the unital map M α ( X ) A ∋ f ( f ( t ) , f (0)) ∈ Γ α (Σ X ) . For every [ α ] ∈ K ( X ) ⊗ Z n ⊂ [ X, Aut( O n +1 )], we have a unitary U α ∈ U ( C ( X ) ⊗ E n +1 ) with π ( U α ) = u ( α ) ∈ C ( X ) ⊗ O n +1 , where π : C ( X ) ⊗ E n +1 → C ( X ) ⊗ O n +1 isthe quotient map. We need the following lemma in Section 3. Lemma 2.10 ([13]) . Let X be a connected finite CW-complex, and let α, U α be as above.Let v ∈ U ( C ( X ) ⊗ M n ( O ∞ )) be a unitary satisfying [ v ] = [ U α ] ∈ K ( C ( X ) ⊗ E n +1 ⊗ M n ( O ∞ )) . Then, we have [1 M α ⊗ Ad v ( X ) ( O n +1 ⊗ M n ( O∞ )) ] = 0 ∈ K ( M α ⊗ Ad v ( X ) ( O n +1 ⊗ M n ( O ∞ )) ) . In particular, one has [1 (Γ α (Σ X ) O n +1 ⊗ C (Σ X ) Γ Ad v (Σ X ) M n ( O∞ ) ) ] = 0 .Proof. Let W be the following unitary W := (cid:18) O n +1 S ∗ S (cid:19) ∈ M n +2 ( O n +1 ) ⊂ C ( X ) ⊗ O n +1 ⊗ M n +2 ( O ∞ ) , where we write S := ( S , · · · , S n +1 ). The unitary W is self-adjoint and there is a con-tinuous path of unitary { V t } t ∈ [0 , ⊂ C ( X ) ⊗ O n +1 ⊗ M n +2 ( O ∞ ) satisfying V = W, V = α ⊗ Ad( v ⊕ )( W ). The unitary V is an element of M α ⊗ Ad( v ⊕ ) ( X ) ( O n +1 ⊗ M n +2 ( O ∞ )) . Itis easy to check that the following inclusion gives isomorphisms of K-groups M α ⊗ Ad v ( X ) ( O n +1 ⊗ M n ( O ∞ )) ∋ f f ⊕ ∈ M α ⊗ Ad( v ⊕ ) ( X ) ( O n +1 ⊗ M n +2 ( O ∞ )) , and one has[1 M α ⊗ Ad v ( X ) ( O n +1 ⊗ M n ( O∞ )) ] = (cid:20)(cid:18) n +1
00 0 (cid:19)(cid:21) − (cid:20)(cid:18) O n +1
00 1 (cid:19)(cid:21) . One has V (cid:18) n +1
00 0 (cid:19) V ∗ = V W (cid:18) O n +1
00 1 (cid:19) W V ∗ , and the direct computation yields V W = (cid:18) v ⊕
00 1 (cid:19) (cid:18) O n +1 α ( S ) ∗ α ( S ) 0 (cid:19) (cid:18) v ∗ ⊕
00 1 (cid:19) (cid:18) O n +1 S ∗ S (cid:19) = (cid:18) v ⊕ u ( α ) (cid:19) (cid:18) O n +1 S ∗ S (cid:19) (cid:18) v ∗ ⊕ u ( α ∗ ) (cid:19) (cid:18) O n +1 S ∗ S (cid:19) = (cid:18) S ∗ ( S ( v ⊕ ) S ∗ u ( α ) ∗ ) S u ( α ) S ( v ∗ ⊕ ) S ∗ (cid:19) . Using the following exact sequence K ( SC ( X ) ⊗ O n +1 ⊗ M n +2 ( O ∞ )) ֒ → K ( M α ⊗ Ad( v ⊕ ) ( X ) ( O n +1 ⊗ M n +2 ( O ∞ )) ) K ( O n +1 ⊗ M n +2 ( O ∞ )) , one can identify [1 M α ⊗ Ad v ( X ) ( O n +1 ⊗ M n ( O∞ )) ] with[ S ( v ⊕ ) S ∗ u ( α ) ∗ ] = [ v ] − [ π ( U α )] = 0 ∈ K ( C ( X ) ⊗ O n +1 ⊗ M n ( O ∞ )) . Let [ E ] ∈ [ X, BAut( E n +1 ⊗ O ∞ )] (resp. [ B ] ∈ [ X, BAut( M n ( O ∞ ))]) denote the iso-morphism class of a locally trivial continuous C ( X )-algebra E (resp. B ) whose fiberis E n +1 ⊗ O ∞ (resp. M n ( O ∞ )). There is a locally trivial continuous C ( X )-algebra A ⊂ E whose fiber is K ⊗ O ∞ , and we have b O ∞ ([ E ]) := [ A ] ∈ E O ∞ ( X ) (see Theorem 2.5). Theorem 3.1.
Let ( X, x ) be a pointed, path connected, finite CW-complex. For ev-ery [ E ] ∈ [ X, BAut( E n +1 ⊗ O ∞ )] , there is a unique element [ B ] ∈ [ X, BAut( M n ( O ∞ ))] satisfying [ K ⊗ B ] = − [ A ] = − b O ∞ ([ E ]) ∈ ¯ E O ∞ ( X ) , [1 B⊗E ] ∈ Im( K ( B ⊗ C ( X ) A ) K (id ⊗ ι ) −−−−−→ K ( B ⊗ C ( X ) E )) . Here, we denote by ι : A ֒ → E the inclusion map. Lemma 2.6 implies [ B ] = [ p ( A − ) p ], and the second condition is equivalent to[1 B⊗O ] = 0 ∈ K ( B ⊗ C ( X ) O ) . Lemma 3.2.
Fix a continuous field E of E n +1 ⊗ O ∞ . For two continuous fields B , B of M n ( O ∞ ) satisfying two conditions in Theorem 3.1 with respect to E , we have [ B ] = [ B ] .Proof. One can find a C ( X )-linear isomorphism γ : K ⊗B → K ⊗B satisfying [ ev x ◦ γ ( e ⊗ B )] = [ e ⊗ ( B ) x ] = n ∈ K (( K ⊗ B ) x ). Let ι : A ֒ → E be as in Theorem 2.5. Sincethe inclusion K ⊗ O ∞ → E n +1 ⊗ O ∞ gives a map Z − n −−→ Z of K -groups, the preimage of[1 B i ⊗E ] should be a “rank −
1” projection in K ( B i ⊗ C ( X ) A ) = K ( X ). Now one has anelement a i ∈ −
1+ ˜ K ( X ) ⊂ K ( K ⊗ B i ⊗ C ( X ) A ) × which is sent to [ e ⊗ B i ⊗ E ] ∈ K ( K ⊗B i ⊗ C ( X ) E ) by the map K (id K ⊗B i ⊗ ι ). The following commutative diagram and Lemma2.2 show that there is a map α ∈ Aut X ( K ⊗ B ) satisfying [ γ ( e ⊗ B )] = [ α ( e ⊗ B )] , · ( K ( γ ⊗ id)( a ) · a − ) = K ( α ) : K ⊗ B ⊗ C ( X ) A id ⊗ ι / / γ ⊗ id (cid:15) (cid:15) K ⊗ B ⊗ C ( X ) E γ ⊗ id (cid:15) (cid:15) K ⊗ B ⊗ o o γ (cid:15) (cid:15) K ⊗ B ⊗ C ( X ) A id ⊗ ι / / K ⊗ B ⊗ C ( X ) E K ⊗ B . id ⊗ o o Therefore, we have B ∼ = B . Proof of Theorem 3.1.
Let ( E i , A i , B i ) , i = 1 , C ( X )-linear isomorphism φ : E → E (i.e.,[ E ] = [ E ]). We show [ B ] = [ B ]. Since the following diagram commutes, B ⊗ C ( X ) A ⊗ ι / / id ⊗ φ | A (cid:15) (cid:15) B ⊗ C ( X ) E ⊗ φ (cid:15) (cid:15) B ⊗ C ( X ) A ⊗ ι / / B ⊗ C ( X ) E , he pair ( E , A , B ) also satisfies the conditions. Now Lemma 3.2 proves the statement.By Theorem 3.1, the map t X : [ X, BAut( E n +1 ⊗O ∞ )] ∋ [ E ] [ B ] ∈ [ X, BAut( M n ( O ∞ ))]is well-defined. For a base point preserving continuous map f : ( Y, y ) → ( X, x )and a continuous field E over X , one has the pull-back of the continuous field f ∗ E := C ( Y ) ⊗ C ( X ) E with a natural homomorphism E ∋ d ⊗ d ∈ C ( Y ) ⊗ C ( X ) E = f ∗ E . Sinceone has f ∗ ( B ⊗ C ( X ) E ) ∼ = f ∗ B ⊗ C ( Y ) f ∗ E , it is easy to check that the map t X is naturalwith respect to X . Definition 3.3.
Let q : Aut( E n +1 ⊗O ∞ ) → Aut( O n +1 ) be the group homomorphism whichis a weak homotopy equivalence. Let C be the category whose objects are pointed, pathconnected, finite CW-complexes, and morphisms are the base point preserving continuousmaps. Let S be the category of sets with a distinguished element and maps preservingthe distinguished elements. For ( X, x ) ∈ C , we regard ( X, x ) [ X, BAut( O n +1 )] and ( X, x ) [ X, BAut( M n ( O ∞ ))] as contravariant functors from C to S where thedistinguished element is the homotopy class of the constant map. We define a naturaltransformation by T X := t X ◦ (B( q ) ∗ ) − : [ X, BAut( O n +1 )] → [ X, BAut( M n ( O ∞ ))] . Proposition 3.4.
One has T X ([ O ]) = [ B ] if and only if [1 B⊗O ] = 0 ∈ K ( B ⊗ C ( X ) O ) .Proof. Assume [1
B⊗O ] = 0. Since there is an exact sequence A → E → O as in Theorem2.5, we have an element a ∈ K ( B ⊗ C ( X ) A ) which is sent to [1 B⊗E ] ∈ K ( B ⊗ C ( X ) E ).Therefore, the element a is “rank −
1” projection, and [9, Th. 4.2] implies − [ A ] = [ K ⊗B ] ∈ ¯ E O ∞ ( X ) (i.e., T X ([ O ]) = [ B ]). Remark 3.5.
One has [ X, BAut( O n +1 )] = [ X, BAut( O n +1 )] and [ X, BAut( M n ( O ∞ ))] =[ X, BAut( M n ( O ∞ ))] as mentioned in Section 2.1. The map T X is bijective for every ( X, x ) ∈ C if and only if T S k is bijective for every k ≥ by Brown’s representabilitytheorem [3, Lem. 1.5]. Lemma 3.6.
Let SX be the reduced suspension of a connected finite CW-complex ( X, x ) .For a path connected group G , we have a natural group isomorphism [ SX, B G ] → [ X, G ] .Proof. Since G is path connected, one has [ SX, B G ] = [ SX, B G ]. For a CW-complex,the quotient map Σ X → SX is a homotopy equivalence, and the map [ SX, B G ] → [Σ X, B G ] is bijective. By [12, Cor. 8.3], we have a bijection [Σ X, B G ] → [ X, G ], wherethe homotopy class of the map α : X → G is sent to the isomorphism class of the principal G -bundle on Σ X whose clutching function over X is the map α . It is easy to check thatthe composition of the above maps is a group homomorphism.Since two classifying spaces BAut( O n +1 ) and BAut( M n ( O ∞ )) have countable homo-topy groups, there are CW-complexes Y , Y with countably many cells and two weakhomotopy equivalences Y → BAut( O n +1 ), Y → BAut( M n ( O ∞ )) (see [11, p 188]). Corollary 3.7.
The map T SX : [ SX,
BAut( O n +1 )] → [ SX,
BAut( M n ( O ∞ ))] gives a grouphomomorphism [ X, Aut( O n +1 )] → [ X, Aut( M n ( O ∞ ))] , and, for X = S k − , the map T S k :[ S k , BAut( O n +1 )] → [ S k , BAut( M n ( O ∞ )] is identified with id K ( S k − ) ⊗ Z n .Proof. First, we show that the map T SX is a group homomorphism. By Lemma 3.6, itsuffices to construct a map f : Y → Y representing the natural transformation[ SX, Y ] → [ SX,
BAut( O n +1 )] T SX −−−→ [ SX,
BAut( M n ( O ∞ ))] → [ SX, Y ] . ow [3, Lem. 1.7] shows the existence of such a map f .Next, we show that the map T S k is bijective. Fix v ∈ U ( C ( X ) ⊗ M n ( O ∞ )) and[ α ] ∈ [ S k − , Aut( O n +1 )] with [ v ] ⊗ ¯1 = [ U α ] ⊗ ¯1 ∈ K ( S k − ) ⊗ Z n = [ S k − , Aut( O n +1 )]as in Lemma 2.10. Lemma 2.10 and Proposition 3.4 implies T S k ([Γ α (Σ S k − ) O n +1 ]) =[Γ Ad v (Σ S k − ) M n ( O ∞ ) ] (i.e., T S k ([ α ]) = [Ad v ]). Therefore, the map T S k is identified withid : K ( S k − ) ⊗ Z n ∋ [ U α ] ⊗ ¯1 [ v ] ⊗ ¯1 ∈ K ( S k − ) ⊗ Z n . Now we have shown our main theorem.
Theorem 3.8.
For every path connected finite CW-complex X , the map T X is bijective. Corollary 3.9.
We have − b O ∞ = B( η ) ∗ ◦ T X and − Im( b O ∞ ) = Im(B( η ) ∗ ) . The map T SX gives a group isomorphism, and the following diagram commutes : K ( X ) ⊗ Z n / / [ X, Aut( O n +1 )] − δ / / T SX (cid:15) (cid:15) (1 + Tor( K ( X ) , Z n )) ×· − (cid:15) (cid:15) K ( X ) ⊗ Z n Ad / / [ X, Aut( M n ( O ∞ ))] η ∗ / / (1 + Tor( K ( X ) , Z n )) × , where the right vertical map sends an invertible element of the K-theory ring to its inverse. Thanks to Remark 2.4, now we have another proof of the result [5, Th. 5.3].
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