Algebraic K-theory, K-regularity, and T-duality of O ∞ -stable C ∗ -algebras
aa r X i v : . [ m a t h . K T ] M a y ALGEBRAIC K-THEORY, K-REGULARITY, AND T -DUALITY OF O ∞ -STABLE C ∗ -ALGEBRAS SNIGDHAYAN MAHANTA
Dedicated to Professor Marc A. Rieffel on the occasion of his 75th birthday.
Abstract.
We develop an algebraic formalism for topological T -duality. More precisely, weshow that topological T -duality actually induces an isomorphism between noncommutativemotives that in turn implements the well-known isomorphism between twisted K-theories(up to a shift). In order to establish this result we model topological K-theory by algebraicK-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C ∗ -algebra D . Then we show that there is a functorial topological K-theory symmetric spectrumconstruction K topΣ ( − ) on the category of separable C ∗ -algebras, such that K topΣ ( D ) is analgebra over this operad; moreover, K topΣ ( A ˆ ⊗D ) is a module over this algebra. Along the waywe obtain a new symmetric spectra valued functorial model for the (connective) topologicalK-theory of C ∗ -algebras. We also show that O ∞ -stable C ∗ -algebras are K-regular providingevidence for a conjecture of Rosenberg. We conclude with an explicit description of thealgebraic K-theory of ax + b -semigroup C ∗ -algebras coming from number theory and thatof O ∞ -stabilized noncommutative tori. Introduction
Within the category of separable C ∗ -algebras SC ∗ the stable ones, i.e., those A ∈ SC ∗ satisfying A ˆ ⊗ K ∼ = A , play a privileged role. For instance, it is known that they satisfythe Karoubi conjecture and appear very naturally in the context of twisted K-theory. Theresults in this article demonstrate that O ∞ -stable separable C ∗ -algebras, i.e., those A ∈ SC ∗ satisfying A ˆ ⊗O ∞ ∼ = A , deserve a similar prominent status. Moreover, the Cuntz algebra O ∞ is strongly self-absorbing , which has several interesting ramifications.Ever since its inception by Kontsevich [33] the homological mirror symmetry conjecture haspromoted rich interaction between geometry, algebra, and (higher) category theory. Mirrorsymmetry is related to T -duality via the Strominger–Yau–Zaslow conjecture [61] and hencewe believe that it is worthwhile to have an algebraic formalism for T -duality at our disposal.One aspect of this theory is the Bunke–Schick topological T -duality , which has received a lotof attention in the mathematical literature. It is insensitive to subtle geometric structuresbut its mathematical underpinnings are very well understood [8, 7, 9]. One of the objectivesof this article is to develop an algebraic formalism for topological T -duality relating it tothe theory of noncommutative motives [34, 35, 65, 45]. Along the way we obtain severalinteresting applications to algebraic K-theory and K-regularity of C ∗ -algebras. The novelty Mathematics Subject Classification.
Key words and phrases. T -duality, noncommutative motives, operads, K-theory, K-regularity, symmetricspectra.This research was supported by the Deutsche Forschungsgemeinschaft (SFB 878 and SFB 1085), ERCthrough AdG 267079, and the Humboldt Professorship of M. Weiss. f our approach lies in the use of the Cuntz algebra O ∞ and the construction of certainoperadic actions on (twisted) K-theory.Let us briefly describe our main results. Given any C ∗ -algebra A we can functorially asso-ciate its noncommutative motive HPf dg ( A ) with it [41]. As a mathematical object HPf dg ( A ) isa differential graded category that is defined purely algebraically. In Section 1 we show that if A and A ′ are KK-equivalent separable C ∗ -algebras, then HPf dg ( A ˆ ⊗O ∞ ) and HPf dg ( A ′ ˆ ⊗O ∞ )are isomorphic objects in the category of noncommutative motives (cf. Theorem 1.5 andCorollary 1.6). We also show that the nonconnective K-theory of the noncommutative mo-tive HPf dg ( A ˆ ⊗O ∞ ) is naturally isomorphic to the topological K-theory of A (cf. Theorem1.7). It is known that under favourable circumstances topological T -duality can be expressedas a KK-equivalence between two separable C ∗ -algebras [3, 4]. Thus our results show thatin such cases one actually has an isomorphism of noncommutative motives that implementsthe well-known isomorphism (up to a shift) between the twisted K-theories. Since noncom-mutative motives constitute the universal cohomology theory of noncommutative spaces, ourresults demonstrate that topological T -duality implements an isomorphism of universal coho-mology theories. The treatment here is completely algebraic; we model topological K-theoryvia algebraic K-theory following (and somewhat refining) our earlier approach in [41].Let us now explain the significance of O ∞ in this context that made no appearance in[41]. It is a noteworthy example of a strongly self-absorbing C ∗ -algebra [66] with veryinteresting structural properties. Using a result from [13] (see also [31]) one can deduce thatnonconnective algebraic K-theory agrees naturally with topological K-theory for O ∞ -stable C ∗ -algebras. There has been considerable interest in associating symmetric spectra with C ∗ -algebras functorially, whose homotopy groups are the topological K-theory groups. Variousalgebraic K-theory machines (like Waldhausen K-theory) canonically produce symmetricspectra (see Remark 1.2.6 of [28]). Combining this fact with our techniques we obtain a newfunctorial symmmetric spectra valued model for the (connective) topological K-theory of C ∗ -algebras (cf. Theorem 2.4 and Remark 2.5). This algebraic method completely circumventsthe analytical difficulties that one needs to overcome in a direct approach like the one in[29]. Applying the technology of [41] this result could have been deduced without invoking O ∞ . However, we exhibit more algebraic structure on K-theory using our current formalism.Indeed, we construct for every strongly self-absorbing C ∗ -algebra D an E ∞ -operad that wecall the strongly self-absorbing D -operad (cf. Definition 3.2 and Proposition 3.4). Then weshow that there is a symmetric spectra valued model for the topological K-theory of D thatis an algebra over the D -operad; moreover, the topological K-theory symmetric spectrumof any D -stable separable C ∗ -algebra is a module over this algebra (cf. Theorem 3.6 for amore general formulation and also Example 3.7). These operadic structures up to coherenthomotopy in symmetric spectra can be further rectified to strict ones (see, for instance,[44, 24]). For the ∞ -categorical counterparts of related results the readers may refer to [39].Using similar ideas as before we show that A ˆ ⊗O ∞ is K-regular for any C ∗ -algebra A (cf.Theorem 4.1), providing evidence for a conjecture of Rosenberg [55]. We carry out an ex-plicit computation of the algebraic K-theory of ax + b -semigroup C ∗ -algebras associated withnumber rings [17, 37] (cf. Theorem 5.1). Noncommutative tori constitute arguably the mostwidely studied class of noncommutative spaces. Their geometric invariants were studiedextensively by Connes and Rieffel (see, for instance, [10, 11, 52]). We show that the alge-braic K-theory of noncommutative tori are explicitly computable after O ∞ -stabilization (cf. heorem 5.2). Using a powerful result of Rieffel [52], one also obtains a clear understandingof the elements of the algebraic K-theory groups in low degrees (see Remark 5.3). Remark.
Some of the arguments below exploit a cute trick (cf. Lemma 1.4). The rangeof applicability of this trick is much broader than the case explored here (see, for instance,Proposition 1.1.2 of [53]). The author is grateful to D. Enders for pointing out that Proposi-tion 2.2, that uses this trick, can be generalized to all C ∗ -algebras of the form A ˆ ⊗ B with B properly infinite . We encourage the readers to consult [13] for an even more general result. Notations and Conventions:
In the sequel we denote the category of all (resp. separa-ble) C ∗ -algebras by C ∗ (resp. SC ∗ ). We denote by K ( − ) [resp. K( − )] the nonconnectivealgebraic K-theory spectrum [resp. K-theory group] functor and by K top ( − ) [resp. K top ( − )]the (twisted) topological K-theory spectrum [resp. K-theory group] functor on C ∗ . Unlessotherwise stated, all spaces are assumed to be Hausdorff and ˆ ⊗ will denote the maximal C ∗ -tensor product (expect in Example 3.7). Acknowledgements:
The author would like to thank J. Cuntz, D. Enders, G. Horel, J.Lind, and A. Thom for beneficial discussions. The author is also grateful to U. Bunke,N. C. Phillips, and J. Rosenberg for constructive feedback. Careful reviews resulting inseveral corrections suggested by the anonymous referees have helped improve the expositionsignificantly. A part of this project was also supported by the fellowship for research scholarsof the Max Planck Institute for Mathematics, Bonn.1.
Topological T -duality and noncommutative motives For the benefit of the reader we briefly discuss topological T -duality and noncommutativemotives as well as their K-theory before explaining our results.1.1. Topological T -duality. T -duality is an interesting phenomenon is string theory, someof whose mathematical aspects were studied in [2]. We are solely going to focus on axiomatictopological T -duality from [8], which builds upon the earlier work in [2]. Let B be a topo-logical base space. Consider the category of pairs ( E, h ), where π : E → B is a principal S -bundle over B and h ∈ H ( E, Z ). Two such pairs ( E , h ) and ( E , h ) are isomorphic if thereis an isomorphism F : E → E of principal bundles such that F ∗ h = h . Two pairs ( E , h )and ( E , h ) are said to be T -dual if there is a Thom class Th ∈ H ( S ( V ) , Z ) for S ( V ) suchthat h = i ∗ Th and h = i ∗ Th . Here S ( V ) is the sphere bundle of V := E × S C ⊕ E × S C and i k : E k → S ( V ) are the canonical maps for k = 1 ,
2. This definition implies the followingcorrespondence picture: Let π k : E k → B with k = 1 , S -bundles and( E , h ) and ( E , h ) be T -dual pairs. Then there is a commutative diagram E × B E z z tttttttttt q (cid:15) (cid:15) pr $ $ ❏❏❏❏❏❏❏❏❏❏ E π $ $ ❏❏❏❏❏❏❏❏❏❏❏ E π z z ttttttttttt B, (1) uch that pr ∗ ( h ) = pr ∗ ( h ). This basic correspondence picture relates topological T -dualityto cohomological quantization (see, for instance, [48, 59]).In [8] Bunke–Schick showed that the association B
7→ { isom. classes of pairs over B } as afunctor on topological spaces is representable. The representing space E supports a universalpair and any pair on B can be obtained up to isomorphism via a pullback along some map B → E (defined uniquely up to homotopy). Using the explicit construction of the universalobject and the T -dual of the universal pair the authors were able to prove the existenceand uniqueness of T -duality for S -bundles. One of the salient features of T -duality is thefollowing: If ( E , h ) and ( E , h ) are T -dual pairs, then there is an isomorphism of twistedK-theories:(K top ) od ( E , h ) ≃ (K top ) ev ( E , h ) and (K top ) ev ( E , h ) ≃ (K top ) od ( E , h ) . (2)The theory of topological T -duality is not limited to S -bundles. However, for more gen-eral ( Q ni =1 S )-bundles with n > C ∗ -algebras [46]. Moreover, C ∗ -algebras appear quite naturally in thecontext of twisted K-theory [54]. Thus it seems natural to study T -duality via C ∗ -algebrasfrom the outset. The readers may refer to [56] for a survey on the interactions between C ∗ -algebras, K-theory, noncommutative geometry, and T -duality. Some recent results indicatethat T -duality can even be related to Langlands duality [21, 6].1.2. Noncommutative motives.
Much like motives in algebraic geometry serve as thereceptacle for the universal cohomology theory for algebraic varieties (with various inter-esting realization functors), noncommutative motives constitute the universal cohomologytheory for noncommutative spaces in the sense of Kontsevich [35, 34]. The rough idea be-hind its construction is to begin with a reasonable category (actually a model category) ofnoncommutative spaces and then enforce certain properties that are expected of any coho-mology theory, i.e., pass to the suspension stabilization and implement appropriate versionsof Morita invariance and localization. The techniques involved in this process are to someextent informed by both algebraic geometry and homotopy theory. The theory of noncom-mutative motives has interesting applications to K-theory as well as a wide variety of otherareas in mathematics [34, 45]. We briskly review its rudiments following [32, 64].Let k be a field of characteristic zero (for our purposes it is C ). Let DGcat denote thecategory of (small) differential graded (DG) categories over k . In this setting a noncom-mutative space is the same thing as a DG category. There is a model category of modules(and even a subcategory of perfect modules) over a DG category A , whose homotopy cat-egory denoted by D ( A ) is the derived category of A . We refer the readers to section 3.8of [32], where it is explained how a functor between DG categories F : A → B induces arestriction of scalars functor D ( B ) → D ( A ). A morphism of DG categories F : A → B iscalled a derived Morita equivalence or simply a
Morita morphism if the restriction of scalarsfunctor D ( B ) → D ( A ) is an equivalence of triangulated categories. The category DGcat supports a model structure, whose weak equivalences are the Morita morphisms, and itshomotopy category is denoted by
Hmo . In the module category over a DG category one canperform (homotopical) analogues of most of the operations that are present on the modulecategory over a ring. Let rep(
A, B ) ⊂ D ( A op ⊗ L B ) be the full triangulated subcategoryconsisting of those A - B -bimodules X , such that X ( a, − ) ∈ D ( Perf ( B )) for every object a ∈ A . Set Hmo to be the category whose objects are DG categories and whose morphisms re Hmo ( A , B ) := K (rep( A , B )) with composition induced by the tensor product of bimod-ules. There is a functor U A : DGcat → Hmo that should be regarded as the pure motiveassociated with a noncommutative space. The functor U A is identity on objects and sendsa morphism of DG categories to its class in the Grothendieck group of bimodules. Thecategory of pure noncommutative motives Hmo further maps into a triangulated category of mixed noncommutative motive Mot locdg . The composite functor U locdg : DGcat U A → Hmo → Mot locdg has the property that it sends every exact sequence of DG categories to an exact triangle in
Mot locdg . Recall that a diagram of DG categories
A → B → C is called an exact sequence if theinduced sequence D ( A ) → D ( B ) → D ( C ) of triangulated categories is Verdier exact. Theconstruction of Mot locdg uses the theory of Grothendieck derivators that we leave out from thediscussion. In this article we are solely going to focus on the category of pure noncommutativemotives
Hmo . There is a stable ∞ -categorical counterpart of noncommutative motives [1]and our results relating topological T -duality with noncommutative motives also admit ageneralization to the ∞ -categorical setup (see Section 4 of [40]).1.3. Nonconnective K-theory of pure noncommutative motives.
Any pure noncom-mutative motive is merely an object
A ∈
DGcat , since the functor U A : DGcat → Hmo is identity on objects. Using the corepresentability result (see Theorem 6.1 of [65]) thequickest definition of the nonconnective algebraic K-theory spectrum of a DG category A is K ( A ) := R Hom( U locdg ( k ) , U locdg ( A )). Note that the category of mixed noncommutativemotives Mot locdg is canonically enriched over spectra; this is a consequence of the general for-malism of triangulated derivators that is deployed to construct
Mot locdg . The nonconnectivealgebraic K-theory groups of A can be defined as K n ( A ) := Hom( U locdg ( k ) , U locdg ( A )[ − n ]) forall n ∈ Z . By construction the functor K ( − ) (resp. K n ( − )) factors through the category ofpure noncommutative motives Hmo .The nonconnective algebraic K-theory of a DG category A can also be constructed bymore traditional methods that we presently explain. The category of perfect cofibrant DGmodules Perf ( A ) over A admits the structure of a complicial exact category, whose weakequivalences are those maps that are isomorphisms in D ( A ). Recall that a module over A isperfect if and only if it is a compact object in the derived category D ( A ) (see Corollary 3.7of [32]). Applying the Waldhausen K-theory functor K w ( − ) to the complicial exact category Perf ( A ) one gets the connective K-theory spectrum K w ( A ) of the DG category A . From[57] (see also Section 3.2.33 of [58]) one learns that there is a suspension construction Σ( − )of a complicial exact category E , such that there is a natural homotopy equivalence K w ( E ) ∼ → Ω K w (Σ( E )) , (3)provided the triangulated category associated with E is idempotent complete (e.g., E = Perf ( A )). The nonconnective K-theory spectrum of A , denoted by K ( A ), is the spectrumwhose n -th space is K w (Σ n ( Perf ( A ))) and the structure maps are furnished by (3). Anyunital k -algebra R can be viewed as a DG category with one object • , such that End( • ) = R .In this case the above construction applied to E = Perf ( R ) recovers Quillen’s higher algebraicK-theory of R in nonnegative degrees and Bass’ negative K-theory in negative degrees. .4. Our results.
We denote the category of separable C ∗ -algebras by SC ∗ and the bivariantK-theory category by KK . There is a canonical functor ι : SC ∗ → KK , which is identity onobjects and admits a universal characterization [26, 16]. Building upon an earlier work ofQuillen [49] the author constructed a functorial passage HPf dg from separable C ∗ -algebrasto (pure) noncommutative motives. For any A ∈ SC ∗ the DG category HPf dg ( A ) consists ofcochain complexes X of right ˜ A -modules, where ˜ A is the unitization of A , that satisfy(1) X is homotopy equivalent to a strictly perfect complex (of right ˜ A -modules), and(2) the quotient complex X/XA is acyclic.The association A HPf dg ( A ) gives rise to a functor SC ∗ → Hmo in an evident manner. Thefollowing two results (amongst others) are proved in [41]: Theorem 1.1.
There is a dashed functor below making the following diagram of categoriescommute (up to a natural isomorphism): SC ∗ A A ˆ ⊗ K / / ι (cid:15) (cid:15) SC ∗ HPf dg (cid:15) (cid:15) KK / / ❴❴❴❴❴❴ Hmo . Theorem 1.2.
For any A ∈ SC ∗ the homotopy groups of the nonconnective K-theory spec-trum of HPf dg ( A ˆ ⊗ K ) are naturally isomorphic to the topological K-theory groups of A . Remark 1.3.
In [41] the author phrased the results in terms of
NCC dg , which was called thecategory of noncommutative DG correspondences. The category NCC dg is equivalent to Hmo .Moreover, in Theorem 3.7 of [41] actually the connective version of Theorem 1.2 was proven.The translation to the nonconnective version based on the above discusion (see subsection1.3) is straightforward. Note that the functor A π ∗ ( K ( HPf dg ( A ˆ ⊗ K ))) is C ∗ -stable, half-exact, and homotopy invariant whence it is Bott 2-periodic and the natural comparisonmap (see Section 2) gives rise to a natural transformation π ∗ ( K ( HPf dg ( A ˆ ⊗ K ))) → K top ∗ ( A )between two 2-periodic homology theories on SC ∗ .A crucial insight of Rosenberg in [54] is that certain bundles of compact operators K on locallycompact spaces can be used to model twisted K-theory that was introduced in [23]. Moreprecisely, given any pair ( E, h ) with E locally compact one can construct a noncommutativestable C ∗ -algebra CT( E, h ), whose topological K-theory is the twisted K-theory of the pair(
E, h ). This formalism extends to certain infinite dimensional spaces through the use of σ - C ∗ -algebras [43]. In [3, 4] the authors extended the formalism of T -duality to C ∗ -algebras andshowed that under favourable circumstances if B and B ′ are T -dual C ∗ -algebras, then thereis an invertible element in KK ( B, Σ B ′ ) that implements the twisted K-theory isomorphism(as in (2)). The Connes–Skandalis picture of KK-theory [12] and Rieffel’s imprimitivityresult [50] are pertinent to their construction. Thanks to Theorem 1.1 we conclude that iftwo stable C ∗ -algebras B and B ′ are T -dual, such that there is an invertible element α ∈ KK ( B, Σ B ′ ), then their noncommutative motives HPf dg ( B ) and HPf dg ( B ′ ) are isomorphicin Hmo . Furthermore, Theorem 1.2 asserts that the invertible element α implements thetwisted K-theory isomorphism (like (2)) that is expected from T -duality.Recall that the Cuntz algebra O ∞ is the universal unital C ∗ -algebra generated by a set ofisometries { s i | i ∈ N } with mutually orthogonal range projections s i s ∗ i [15]. Observe that O ∞ s a unital C ∗ -algebra, so that O ∞ -stabilization preserves unitality (unlike K -stabilization).The following Lemma is crucial and it exploits the fact that O ∞ is purely infinite . Lemma 1.4.
There is a commutative diagram in C ∗ O ∞ ι / / θ $ $ ■■■■■■■■■ O ∞ O ∞ ˆ ⊗ K , κ : : ✉✉✉✉✉✉✉✉✉ (4)where the top horizontal arrow ι : O ∞ → O ∞ is an inner endomorphism. Proof.
Observe that the subset { s i s ∗ j | i, j ∈ N } ⊂ O ∞ generates a copy of the compactoperators K inside O ∞ . Consider the ∗ -homomorphism κ : O ∞ ˆ ⊗ K → O ∞ , which is definedas a ⊗ e ij s i as ∗ j . Due to the simplicity of all the C ∗ -algebras in sight, κ is injective. Let θ : O ∞ → O ∞ ˆ ⊗ K be simply the corner embedding, sending a a ⊗ e . The composite ι = κθ is given by ι ( a ) = s as ∗ . This ∗ -homomorphism is manifestly inner. (cid:3) Recall that a functor F : SC ∗ → Hmo is called split exact if it sends a split exact sequencein SC ∗ to a direct sum diagram in the additive category Hmo . It follows from Lemma 3.1 of[41] that the functor HPf dg ( − ) is split exact. Theorem 1.5. If A and A ′ are isomorphic in KK , then the noncommutative motives of A ˆ ⊗O ∞ and A ′ ˆ ⊗O ∞ are isomorphic in Hmo . Proof.
Let us first assume that
A, A ′ are unital and let α ∈ KK ( A, A ′ ) be any invertibleelement. Consider the commutative diagram that is obtained by applying A ˆ ⊗− to thecommutative diagram 4 A ˆ ⊗O ∞ id A ˆ ⊗ ι / / R :=id A ˆ ⊗ θ & & ◆◆◆◆◆◆◆◆◆◆ A ˆ ⊗O ∞ A ˆ ⊗O ∞ ˆ ⊗ K . S :=id A ˆ ⊗ κ ♣♣♣♣♣♣♣♣♣♣ (5)Now from Theorem 1.1 one obtains a diagram in Hmo HPf dg ( A ˆ ⊗O ∞ ) HPf dg ( R ) / / HPf dg ( A ˆ ⊗O ∞ ˆ ⊗ K ) HPf dg ( S ) / / β = HPf dg ( α ˆ ⊗ id O∞ ˆ ⊗ id K ) (cid:15) (cid:15) HPf dg ( A ˆ ⊗O ∞ ) HPf dg ( A ′ ˆ ⊗O ∞ ) HPf dg ( R ′ ) / / HPf dg ( A ′ ˆ ⊗O ∞ ˆ ⊗ K ) HPf dg ( S ′ ) / / HPf dg ( A ′ ˆ ⊗O ∞ ) . (6)where R ′ and S ′ are defined in the obvious manner (replace A by A ′ in diagram 5). Since α is invertible, so are α ˆ ⊗ id O ∞ and α ˆ ⊗ id O ∞ ˆ ⊗ id K . Therefore, the middle vertical arrow β is an isomorphism. Observe that S ◦ R is an inner endomorphism in SC ∗ of the form x ( A ⊗ s ) x ( A ⊗ s ) ∗ (and so is S ′ ◦ R ′ similarly). It is known that if F is a matrix stablefunctor on C ∗ (resp. SC ∗ ) and f is an inner endomorphism in C ∗ (resp. SC ∗ ), then F ( f ) isthe identity map (see, for instance, Proposition 3.16. of [19]). It was shown in Lemma 2.3of [41] that the functor HPf dg ( − ) is matrix stable on SC ∗ , whence we get HPf dg ( S ) ◦ HPf dg ( R ) = id HPf dg ( A ˆ ⊗O ∞ ) and HPf dg ( S ′ ) ◦ HPf dg ( R ′ ) = id HPf dg ( A ′ ˆ ⊗O ∞ ) . hus the maps HPf dg ( R ) and HPf dg ( R ′ ) possess left inverses. An inspection of diagram6 reveals that it suffices to show that they also possess right inverses. The composite ∗ -homomorphism K i ֒ → O ∞ θ → O ∞ ˆ ⊗ K defines an invertible element θ ◦ i = γ ∈ KK ( K , O ∞ ˆ ⊗ K ).Consequently, id A ˆ ⊗ γ ∈ KK ( A ˆ ⊗ K , A ˆ ⊗O ∞ ˆ ⊗ K ) is an invertible element. By Theorem 1.1id A ˆ ⊗ γ ˆ ⊗ id K = (id A ˆ ⊗ θ ˆ ⊗ id K ) ◦ (id A ˆ ⊗ i ˆ ⊗ id K ) induces an isomorphism HPf dg ( A ˆ ⊗ K ˆ ⊗ K ) ∼ → HPf dg ( A ˆ ⊗O ∞ ˆ ⊗ K ˆ ⊗ K ) . Let us set I = id A ˆ ⊗ i , so that HPf dg ( R ˆ ⊗ id K ) ◦ HPf dg ( I ˆ ⊗ id K ) is the above isomorphism. Nowconsider the following commutative diagram A ˆ ⊗ K / / I (cid:15) (cid:15) A ˆ ⊗ K ˆ ⊗ K I ˆ ⊗ id K (cid:15) (cid:15) A ˆ ⊗O ∞ / / R (cid:15) (cid:15) A ˆ ⊗O ∞ ˆ ⊗ K R ˆ ⊗ id K (cid:15) (cid:15) A ˆ ⊗O ∞ ˆ ⊗ K / / A ˆ ⊗O ∞ ˆ ⊗ K ˆ ⊗ K . Here all the horizontal arrows are corner embeddings. Now the top and the bottom horizontalarrows are homotopic to isomorphisms. Since
HPf dg ( − ) is homotopy invariant on stable C ∗ -algebras, it sends the top and the bottom horizontal arrows to isomorphisms. We alreadyknow that it sends ( R ˆ ⊗ id K ) ◦ ( I ˆ ⊗ id K ) to an isomorphism. It follows that HPf dg ( R ) has aright inverse. Similarly, one can prove that HPf dg ( R ′ ) has a right inverse. Now using splitexactness of HPf dg ( − ) one can extend the result to nonunital C ∗ -algebras. (cid:3) Corollary 1.6.
The functor
HPf dg ( − ˆ ⊗O ∞ ) is C ∗ -stable and it factors through KK . Proof.
For any separable C ∗ -algebra A the corner embedding A → A ˆ ⊗ K is KK -invertiblewhence HPf dg ( − ˆ ⊗O ∞ ) is C ∗ -stable. It follows from Lemma 3.1 of [41] that the functor HPf dg ( − ˆ ⊗O ∞ ) is split exact. The second assertion now is a consequence of the universalcharacterization of KK . (cid:3) Now we prove the O ∞ -analogue of Theorem 1.2. Theorem 1.7.
For any A ∈ SC ∗ the homotopy groups of the nonconnective K-theory spec-trum of HPf dg ( A ˆ ⊗O ∞ ) are naturally isomorphic to the topological K-theory groups of A . Proof.
By the above Corollary the nonconnective K-theory spectra of
HPf dg ( A ˆ ⊗O ∞ ) and HPf dg ( A ˆ ⊗O ∞ ˆ ⊗ K ) are weakly equivalent. By Theorem 1.2 the homotopy groups of thenonconnective K-theory spectrum of HPf dg ( A ˆ ⊗O ∞ ˆ ⊗ K ) are isomorphic to the topologicalK-theory groups of A ˆ ⊗O ∞ , which are in turn isomorphic to those of A . (cid:3) Remark 1.8 (Categorification of topological T -duality) . It is shown in Example 5.6 of[4] that if two pairs ( E , h ) and ( E , h ) over B with are T -dual, then one can con-struct an invertible element in KK (CT( E , h ) , CT( E , h )). It follows that CT( E , h )and ΣCT( E , h ) are isomorphic in KK whence by Theorem 1.5 HPf dg (CT( E , h ) ˆ ⊗O ∞ ) and HPf dg (ΣCT( E , h ) ˆ ⊗O ∞ ) are isomorphic in Hmo . Now using Theorem 1.7 one concludesK ∗ ( HPf dg (CT( E i , h i ) ˆ ⊗O ∞ )) ∼ = K top ∗ (CT( E i , h i )) ∼ = (K top ) ∗ ( E i , h i ) or i = 1 , T -duality. Since noncommutative motives constitute the universal additiveinvariant [64], an isomorphism therein is the most fundamental (co)homological isomorphism. Remark 1.9.
The introduction of O ∞ in this setting is quite interesting because of theDixmier–Douady theory via O ∞ ˆ ⊗ K -bundles due to Dadarlat–Pennig [20] that is able to see higher twists of K top -theory. This avenue of research deserves further attention.2. The generalized homology theory K ( − ˆ ⊗O ∞ )Let hSp denote the triangulated stable homotopy category. A functor F : C ∗ → hSp is called homotopy invariant if it sends the evaluation at t map ev t : A [0 , → A to anisomorphism in hSp for all A ∈ C ∗ . Such a functor is called excisive if for any short exactsequence 0 → A → B → C → C ∗ the induced diagram F ( A ) → F ( B ) → F ( C ) → Σ F ( A ) is an exact triangle in hSp . A homotopy invariant excisive functor F : C ∗ → hSp is called an hSp -valued generalized homology theory on C ∗ . It is known that the algebraicK-theory functor K ( − ) acquires special properties after stabilization with respect to thecompact operators. We are going to show that the same is true after O ∞ -stabilization. Proposition 2.1.
The functor K ( − ˆ ⊗O ∞ ) : C ∗ → hSp is an excisive functor. Proof.
It follows from the Suslin–Wodzicki Theorem [62, 63] that the functor K is excisiveon C ∗ . Since maximal C ∗ -tensor product is exact, the functor − ˆ ⊗O ∞ preserves exactness in C ∗ whence K ( − ˆ ⊗O ∞ ) is excisive. (cid:3) Thanks to the Karoubi conjecture, which is now a Theorem [62, 63], we know that thenonconnective algebraic K-theory of a stable C ∗ -algebra is isomorphic to its topological K-theory. In fact, there is a canonical comparison map of spectra that induces the isomorphisms c n ( A ) : K n ( A ) → K top n ( A ) for all n ∈ Z when A is stable [30] (see also [55]). The comparisonmap c ( A ) : K ( A ) → K top0 ( A ) is always an isomorphism. Proposition 2.2 (Corti˜nas–Phillips, Karoubi–Wodzicki) . For any C ∗ -algebra A the com-parison map c n ( A ˆ ⊗O ∞ ) : K n ( A ˆ ⊗O ∞ ) → K top n ( A ˆ ⊗O ∞ ) is an isomorphism for all n ∈ Z . Proof.
Let us first assume that A is a unital C ∗ -algebra. After applying A ˆ ⊗− to the com-mutative diagram 4 in Lemma 1.4 we obtain A ˆ ⊗O ∞ id ˆ ⊗ ι / / R :=id ˆ ⊗ θ & & ▼▼▼▼▼▼▼▼▼▼▼ A ˆ ⊗O ∞ A ˆ ⊗O ∞ ˆ ⊗ K , S :=id ˆ ⊗ κ qqqqqqqqqqq (7)where the top horizontal arrow is an inner endomorphism. Now applying the functors K n ( − ),K top n ( − ) and using the naturality of c n , we get a commutative diagramK n ( A ˆ ⊗O ∞ ) K n ( R ) / / c n ( A ˆ ⊗O ∞ ) (cid:15) (cid:15) K n ( A ˆ ⊗O ∞ ˆ ⊗ K ) K n ( S ) / / c n ( A ˆ ⊗O ∞ ˆ ⊗ K ) (cid:15) (cid:15) K n ( A ˆ ⊗O ∞ ) c n ( A ˆ ⊗O ∞ ) (cid:15) (cid:15) K top n ( A ˆ ⊗O ∞ ) K top n ( R ) / / K top n ( A ˆ ⊗O ∞ ˆ ⊗ K ) K top n ( S ) / / K top n ( A ˆ ⊗O ∞ ) . (8) ince S ◦ R is the inner endomorphism id ˆ ⊗ ι : A ˆ ⊗O ∞ → A ˆ ⊗O ∞ , we conclude that K n ( S ) ◦ K n ( R ) is the identity map due to the matrix stability of algebraic K-theory on the cate-gory of unital C ∗ -algebras. Moreover, K top n ( S ) ◦ K top n ( R ) is also the identity map due to thematrix stability of K top n ( − ). The assertion for unital A now follows by a simple diagramchase. Indeed, it is easily seen that K n ( R ) must be injective and K top n ( S ) must be surjec-tive. Since A ˆ ⊗O ∞ ˆ ⊗ K is stable, we conclude that c n ( A ˆ ⊗O ∞ ˆ ⊗ K ) is an isomorphism. Thus c n ( A ˆ ⊗O ∞ ˆ ⊗ K ) ◦ K n ( R ) is injective whence so is c n ( A ˆ ⊗O ∞ ) (the left vertical one). Similarly,K top n ( S ) ◦ c n ( A ˆ ⊗O ∞ ˆ ⊗ K ) is surjective whence so is c n ( A ˆ ⊗O ∞ ) (the right vertical one).The proof for nonunital A follows by a simple excision argument (see Proposition 2.1). (cid:3) Remark 2.3.
The above Proposition also follows from a result of Corti˜nas–Phillips. Theyproved that the comparison map c n ( A ) : K n ( A ) → K top n ( A ) is an isomorphism in a moregeneral setting [13]. Our argument above is based on a strategy of Karoubi–Wodzicki [31]with some simplifications that exploit the special properties of O ∞ . We have decided toinclude our simple proof as it involves only elementary (homological) algebra and hence it is(hopefully) comprehensible to non-experts on C ∗ -algebras.Recall from [29] (see also Section 8.3 of [19]) that for any C ∗ -algebra A there is a functorialspectrum K top ( A ), whose homotopy groups are the topological K-theory groups of A . Theorem 2.4.
For every A ∈ C ∗ there is a natural isomorphism K ( A ˆ ⊗O ∞ ) ∼ = K top ( A ) in hSp , i.e., the functor K ( − ˆ ⊗O ∞ ) : C ∗ → hSp is a model for topological K-theory. Proof.
It follows from Proposition 2.2 that the natural comparison map of spectra is a weakequivalence. Since the canonical ∗ -homomorphism A → A ˆ ⊗O ∞ sending a a ⊗ O ∞ is aKK-equivalence, we have a zigzag of weak equivalences of spectra K ( A ˆ ⊗O ∞ ) ∼ → K top ( A ˆ ⊗O ∞ ) ∼ ← K top ( A ) . Thus for every A ∈ C ∗ there is a natural isomorphism K ( A ˆ ⊗O ∞ ) ∼ = K top ( A ) in hSp . (cid:3) Remark 2.5.
It is interesting to associate with any C ∗ -algebra a symmetric spectrum,whose homotopy groups are the topological K-theory groups. One strategy that circumventsthe intricacies involved in constructing symmetric spectra of topological K-theory directly isthe following: for any unital C ∗ -algebra A construct the (connective) Waldhausen K-theory[67] of A ˆ ⊗O ∞ ; it produces naturally a symmetric spectrum [25] and thanks to Theorem 2.4it is a model for the (connective) topological K-theory of A . The Green–Julg–RosenbergTheorem also enables us to treat the G -equivariant case for a finite group G . Indeed, onecan simply apply the (connective) Waldhausen K-theory functor to ( A ⋊ G ) ˆ ⊗O ∞ for anyunital G - C ∗ -algebra A and use the natural identificationsK ∗ (( A ⋊ G ) ˆ ⊗O ∞ ) ∼ = K top ∗ (( A ⋊ G ) ˆ ⊗O ∞ ) ∼ = K top ∗ ( A ⋊ G ) ∼ = (K top ) G ∗ ( A ) . Using Theorem 1.7 we can handle the situation if A is nonunital or G is not finite (butcompact) or both. This construction would radically differ from that of [29].3. Strongly self-absorbing operads
Operadic structures have pervaded many areas of mathematics and physics with wideranging applications. From the viewpoint of topology the operadic machinery can be effec-tively used to recognise (infinite) loop spaces. An operad in a symmetric monoidal category C , ⊗ , C ) consists of a collection of objects { C ( j ) } j > with each C ( j ) carrying a right actionof the permutation group Σ j , a unit map η : C → C (1), and product or composition maps γ = γ j , ··· ,j k : C ( k ) ⊗ C ( j ) ⊗ · · · ⊗ C ( j k ) → C ( j )for k > j s > s = 1 , · · · , k subject to j = P s j s . These data should be inter-compatible in a specific manner, i.e., satisfy certain associativity, unitality, and equivarianceaxioms (see, for instance, [36]). Neglecting the actions of the permutation groups and thecorresponding equivariance conditions one arrives at the notion of a nonsymmetric operad .In this section a space is tacitly assumed to be compactly generated and weakly Hausdorff.Observe that such spaces constitute a symmetric monoidal category under cartesian productwith pt (a singleton space) as a unit object and hence one may consider operads in spaces.Recall that a unital separable C ∗ -algebra D ( D 6 = C ) is called strongly self-absorbing ifthere is an isomorphism D ∼ → D ˆ ⊗D that is approximately unitarily equivalent to the firstfactor embedding D → D ˆ ⊗D sending d d ⊗ D [66]. Such C ∗ -algebras turn out to besimple and nuclear. The Cuntz algebra O ∞ is a prominent example of such a C ∗ -algebra.For any strongly self-absorbing C ∗ -algebra D we set D ( j ) = Hom ( D ˆ ⊗ j , D ), i.e., the space ofunital full ∗ -homomorphisms D ˆ ⊗ j → D with the point-norm topology. Since D is a separable C ∗ -algebra, it follows from Lemma 22 of [47] that each D ( j ) is a metrizable topological space.Hence they are all compactly generated and Hausdorff spaces. Lemma 3.1.
The collection {D ( j ) } j > can be promoted to an operad in spaces. Proof.
Let us define γ and η as follows: γ : D ( k ) × D ( j ) × · · · × D ( j k ) → D ( j ) = D ( j + · · · + j k )( α, β , · · · , β k ) α ◦ ( β ⊗ · · · ⊗ β k )and η : pt → D (1) sends the unique element in pt to id : D → D . If we let the permutationgroup Σ j act on D ( j ) = Hom ( D ˆ ⊗ j , D ) by permuting the tensor factors of D ˆ ⊗ j , then it canbe verified that the data satisfy the associativity, unitality, and equivariance axioms. (cid:3) Thanks to the above Lemma we introduce the following operad:
Definition 3.2.
For any strongly self-absorbing C ∗ -algebra D we call the operad that thecollection {D ( j ) } j > defines as the strongly self-absorbing D -operad . Remark 3.3.
For j = 0 we get D (0) = pt, i.e., a singleton set containing the unique unitalinclusion C ֒ → D . Hence the strongly self-absorbing D -operad is a reduced operad . Proposition 3.4.
Every strongly self-absorbing D -operad is an E ∞ -operad. Proof.
We need to show that each D ( j ) for j > j on each D ( j ) is free. The contractibility of each D ( j ) follows from Theorem 2.3 of [20] and the factthat D ∼ = D ˆ ⊗ j . In order to see the freeness of the Σ j -action on D ( j ) we check the stabilizers.For any f ∈ D ( j ) suppose f σ = f . Owing to the simplicity of D ˆ ⊗ n any such f ∈ D ( j ) mustbe a monomorphism whence σ has to be the trivial permutation. (cid:3) Remark 3.5.
The underlying nonsymmetric operad of every strongly self-absorbing D -operad is an A ∞ -operad. functor G : ( E , ⊗ E , E ) → ( F , ⊗ F , F ) between symmetric monoidal categories is called lax symmetric monoidal if there is a morphism F → G ( E ) in F and natural transformations κ : G ( A ) ⊗ F G ( B ) → G ( A ⊗ E B )(9)for all A, B ∈ E that satisfy certain well-known associativity, unitality, and symmetry condi-tions. If E and F are (pointed) topological categories, i.e., they are enriched over (pointed)spaces, then a functor G : E → F is called enriched if for all
A, B ∈ E the induced mapMap E ( A, B ) → Map F ( G ( A ) , G ( B )) is (pointed) continuous. Here we have adopted theconvention that in the enriched setting we denote the (pointed) space of morphisms byMap E ( − , − ), Map F ( − , − ), and so on. A symmetric monoidal pointed topological category B = ( B , ∧ , B ) is said to be equipped with a closed action of pointed spaces S ∗ if, for every X ∈ B and for every pair of pointed spaces K, L , the following hold: • the functor ( − ) ∧ X : S ∗ → B is the (enriched) left adjoint of Map B ( X, − ) : B → S ∗ , • the functor K ∧ ( − ) : B → B admits an (enriched) right adjoint ( − ) K : B → B , and • there are coherent natural isomorphisms ( K ∧ L ) ∧ X ∼ = K ∧ ( L ∧ X ) and S ∧ X ∼ = X .Let C be any operad in spaces. An object X ∈ B is said to be an algebra over C if there aremaps θ : C ( j ) + ∧ X ∧ j → X in B for all j >
0, that are associative, unital, and equivariantin a suitable sense [36]. Moreover, an object M ∈ B is said to be a module over X if thereare maps λ : C ( j ) + ∧ X ∧ ( j − ∧ M → M in B for all j > X as an algebra object in B over C via a morphism of topological operads C + → End B ( X ), where End B ( X ) denotes theendomorphism operad of X in B (so that End B ( X )( j ) = Map B ( X ∧ j , X )).A typical example for B that the reader should keep in mind is the category of symmetricspectra Sp Σ . It has an associative and commutative smash product ∧ (with the spherespectrum S as the unit object) that is accompanied by a well developed theory of rings andmodules. The category of symmetric spectra in pointed spaces is enriched, tensored andcotensored over S ∗ , i.e., it admits a closed action of S ∗ (see Propositions 1.3.1 and 1.3.2 of[28] and Example 3.36 of [60] for their topological counterparts). Therefore, the symmetricmonoidal category ( Sp Σ , ∧ , S ) satisfies the assumptions on ( B , ∧ , B ) mentioned above. It isalso a stable model category, whose homotopy category is equivalent to hSp . For furtherdetails consult [28, 60].Recall that ( C ∗ , ˆ ⊗ , C ) is a symmetric monoidal category, where ˆ ⊗ is the maximal (orthe minimal) C ∗ -tensor product. It is also enriched over (pointed) spaces if we endow themorphism sets with the point-norm topology. Let C denote a symmetric monoidal topologicalsubcategory of that of C ∗ -algebras that contains all strongly self-absorbing C ∗ -algebras, e.g., C = SC ∗ or the category of nuclear separable C ∗ -algebras. In each case ˆ ⊗ can be either themaximal or the minimal C ∗ -tensor product. Theorem 3.6.
Let D be any strongly self-absorbing C ∗ -algebra and ( B , ∧ , B ) be any sym-metric monoidal pointed topological category that is equipped with a closed action of pointedspaces. Let C be a symmetric monoidal topological subcategory of C ∗ as above and F : C → B be a lax symmetric monoidal enriched functor. Then F ( D ) is an algebra object in B over thestrongly self-absorbing D -operad. Moreover, for any A ∈ C , the object F ( D ˆ ⊗ A ) ∼ = F ( A ˆ ⊗D )is a module over F ( D ). roof. We define for all j > θ : D ( j ) + ∧ F ( D ) ∧ j → F ( D )( f, ( x , · · · , x j )) f ∗ ( κ ( x , · · · , x j )) . Here f ∗ : F ( D ˆ ⊗ j ) → F ( D ) is the map induced by f ∈ D ( j ) and κ : F ( D ) ∧ j → F ( D ˆ ⊗ j ) is thecanonical map induced by (9). Similarly, we define for all j > λ : D ( j ) + ∧ F ( D ) ∧ ( j − ∧ F ( D ˆ ⊗ A ) → F ( D ˆ ⊗ A )( f, ( x , · · · , x j − ) , y ) ( f ⊗ id) ∗ ( κ ′ (( x , · · · , x j ) , y )) . Here κ ′ : F ( D ) ∧ ( j − ∧ F ( D ˆ ⊗ A ) → F ( D ˆ ⊗ ( j − ) ∧ F ( D ˆ ⊗ A ) → F ( D ˆ ⊗ j ˆ ⊗ A ) is induced by thecomposition of the canonical maps furnished by (9) and ( f ⊗ id) ∗ : F ( D ˆ ⊗ j ˆ ⊗ A ) → F ( D ˆ ⊗ A )is the map induced by f ⊗ id : D ˆ ⊗ j ˆ ⊗ A → D ˆ ⊗ A .We claim that θ and λ are morphisms in B . The enriched functor F gives a continuous map D ( j ) + → Map B ( F ( D ˆ ⊗ j ) , F ( D )). There is also a continuous map Map B ( F ( D ˆ ⊗ j ) , F ( D )) → Map B ( F ( D ) ∧ j , F ( D )) induced by κ : F ( D ) ∧ j → F ( D ˆ ⊗ j ). The composite continuous map D ( j ) + → Map B ( F ( D ) ∧ j , F ( D )) translates to a map D ( j ) + ∧ F ( D ) ∧ j → F ( D ) in B via theclosed action of S ∗ on B , which is seen to be θ . Similar arguments show that λ is also a mapin B . The axiom for unitality says that the following diagrams commute: S ∧ F ( D ) η ∧ id (cid:15) (cid:15) ∼ = / / F ( D ) S ∧ F ( D ˆ ⊗ A ) η ∧ id (cid:15) (cid:15) ∼ = / / F ( D ˆ ⊗ A ) D (1) + ∧ F ( D ) θ ♣♣♣♣♣♣♣♣♣♣♣ D (1) + ∧ F ( D ˆ ⊗ A ) . λ ♠♠♠♠♠♠♠♠♠♠♠♠♠ This condition is clear from the fact that η maps the non-basepoint in S to id : D → D (seeLemma 3.1). Now using the hypothesis that F ( − ) is a lax symmetric monoidal functor onecan check the required associativity and equivariance conditions. (cid:3) Example 3.7.
We demonstrate the utility of our result with an important and interestingexample. There is a construction of the topological K-theory spectrum K topΣ ( − ) of a separable C ∗ -algebra with values in symmetric spectra Sp Σ that satisfies the hypotheses of the aboveTheorem (see Lemma 3.4 of [22]). Note that the authors work in the symmetric monoidalcategory of Z / C ∗ -algebras SC ∗ Z / and use the graded minimal C ∗ -tensorproduct for the symmetric monoidal structure thereon. Hence we equip SC ∗ with the minimal C ∗ -tensor product and consider the (lax) symmetric monoidal enriched functor SC ∗ → SC ∗ Z / that endows any A ∈ SC ∗ with the trivial Z / K topΣ : SC ∗ Z / → Sp Σ works even for separable C ∗ -algebras equipped with an additional compact group action. Itis a variation of the construction in [5] that landed in orthogonal spectra. The enrichmentof the composite functor SC ∗ → SC ∗ Z / K topΣ → Sp Σ follows from the fact that SC ∗ and SC ∗ Z / arethemselves enriched over (pointed) spaces and K topΣ ( − ) is defined as a symmetric sequenceof mapping spaces in SC ∗ Z / (see Section 3.1 of [22]). Remark 3.8.
Thanks to Proposition 3.4 and Theorem 3.6, for any strongly self-absorbing C ∗ -algebra D , one might call K topΣ ( D ) an E ∞ -algebra object in symmetric spectra. UsingRemark 0.14 of [44] (see also Theorem 1.4 of [24]) one can rectify this E ∞ -algebra structure n K topΣ ( D ) (resp. the module structure on K topΣ ( D ˆ ⊗ A )) to a strictly commutative algebrastructure in Sp Σ (resp. to a strict module structure over the latter).4. K -regularity of O ∞ -stable C ∗ -algebras Let F be any functor on C ∗ . A C ∗ -algebra A is called F -regular if the canonical inclusion A → A [ t , · · · , t n ] induces an isomorphism F ( A ) ∼ → F ( A [ t , · · · , t n ]) for all n ∈ N . This maphas a one-sided inverse induced by the evaluation map ev . Rosenberg conjectured that any C ∗ -algebra A is K -regular. Using the techniques developed to prove the Karoubi conjectures[27], it is shown in Theorem 3.4 of [55] that the conjecture is true if A is stable. In fact, theTheorem in [55] asserts that a stable C ∗ -algebra is K m -regular for all m ∈ Z . A C ∗ -algebrais called K -regular if it is K m -regular for all m ∈ Z . Theorem 4.1.
The C ∗ -algebras A ˆ ⊗O ∞ are K-regular for all A ∈ C ∗ . Proof.
In order to avoid notational clutter let us set B [ n ] := B [ t , · · · , t n ] for any B ∈ C ∗ .Using excision we may assume that A is unital. Arguing as in the proof of Proposition 2.2we obtain a commutative diagramK m ( A ˆ ⊗O ∞ ) / / (cid:15) (cid:15) K m ( A ˆ ⊗O ∞ ˆ ⊗ K ) / / (cid:15) (cid:15) K m ( A ˆ ⊗O ∞ ) (cid:15) (cid:15) K m (( A ˆ ⊗O ∞ )[ n ]) / / K m (( A ˆ ⊗O ∞ ˆ ⊗ K )[ n ]) / / K m (( A ˆ ⊗O ∞ )[ n ]) . (10)Due to the stability of A ˆ ⊗O ∞ ˆ ⊗ K the middle vertical arrow is an isomorphism. Moreover,the compositions of the top and the bottom horizontal arrows are again isomorphisms dueto the matrix stability of the functor K m ( − ) for unital algebras. Observe that the composite ∗ -homomorphisms A ˆ ⊗O ∞ → A ˆ ⊗O ∞ ˆ ⊗ K → A ˆ ⊗O ∞ and ( A ˆ ⊗O ∞ )[ n ] → ( A ˆ ⊗O ∞ ˆ ⊗ K )[ n ] → ( A ˆ ⊗O ∞ )[ n ] are still inner. Now a similar diagram chase as before enables one to concludethat the left vertical arrow must be an isomorphism. (cid:3) Remark 4.2.
Purely infinite simple C ∗ -algebras like O ∞ can be regarded as maximallynoncommutative . Rather surprisingly, one needs fairly sophisticated techniques to establishthe K-regularity of commutative C ∗ -algebras (see [55, 14]). Remark 4.3.
Using Proposition 2.2 and Theorem 4.1 the reduction principle for assemblymaps (see Theorem 1.1 of [42]) can be generalized to include O ∞ as coefficients, i.e., fora countable, discrete, and torsion free group G , if the Baum–Connes assembly map withcomplex coefficients is injective (resp. split injective), then the Farrell–Jones assembly mapin algebraic K-theory with O ∞ -coefficients is also injective (resp. split injective).5. Algebraic K -theory of certain O ∞ -stable C ∗ -algebras We now explicitly compute the algebraic K-theory groups of certain O ∞ -stable C ∗ -algebras.It must be noted that complete calculation of the algebraic K-theory groups of an arbitraryring is an extremely difficult task in general. .1. Semigroup C ∗ -algebras coming from number theory. A recent result of Li assertsthat for a countable integral domain R with vanishing Jacobson radical (which is, in addition,not a field) the left regular ax + b -semigroup C ∗ -algebra C ∗ λ ( R ⋊ R × ) is O ∞ -absorbing, i.e., C ∗ λ ( R ⋊ R × ) ˆ ⊗O ∞ ∼ = C ∗ λ ( R ⋊ R × ) (see Theorem 1.3 of [38]). Now we focus on the object ofour interest, namely, the left regular ax + b -semigroup C ∗ -algebra C ∗ λ ( R ⋊ R × ) of the ring ofintegers R of a number field K . It is shown in [18] that in this caseK top ∗ ( C ∗ λ ( R ⋊ R × )) ∼ = ⊕ [ X ] ∈ G \I K top ∗ ( C ∗ ( G X )) , where I is the set of fractional ideal of R , G = K ⋊ K × , and G X is the stabilizer of X underthe G -action on I . The orbit space G \ I can be identified with the ideal class group of K .As a consequence of Proposition 2.2 we obtain Theorem 5.1.
The algebraic K-theory of the ax + b -semigroup C ∗ -algebra of the ring ofintegers R of a number field K is 2-periodic and explicitly given byK ∗ ( C ∗ λ ( R ⋊ R × )) ∼ = ⊕ [ X ] ∈ G \I K top ∗ ( C ∗ ( G X )) . O ∞ -stabilized noncommutative tori. We recall some basic material before statingour result. A good reference for generalities on noncommutative tori is Rieffel’s survey [52].For any real-valued skew bilinear form θ on Z n ( n >
2) the C ∗ -algebra of the noncommutative n -torus A nθ can be defined as the universal C ∗ -algebra generated by unitaries U x ∈ Z n subjectto the relation U x U y = exp( πiθ ( x, y )) U x + y ∀ x, y ∈ Z n . Using the Pimnser–Voiculescu exact sequence one can compute the K top -theory of A nθ as anabelian group, namely,K top0 ( A nθ ) ≃ Z n − and K top1 ( A nθ ) ≃ Z n − . (11) Theorem 5.2.
The algebraic K-theory of the O ∞ -stabilized noncommutative n -torus A nθ is2-periodic and explicitly given byK ( A nθ ˆ ⊗O ∞ ) ≃ Z n − and K ( A nθ ˆ ⊗O ∞ ) ≃ Z n − . Proof.
By Proposition 2.2 one has an isomorphism K ∗ ( A nθ ˆ ⊗O ∞ ) ∼ = K top ∗ ( A nθ ˆ ⊗O ∞ ). Observethat K top0 ( O ∞ ) ≃ Z and K top1 ( O ∞ ) ≃ C ∗ -algebras in sight belong to the UCT-class.Using the K¨unneth Theorem (or possibly O ∞ -stability in K top -theory) one now deduces thatK top ∗ ( A nθ ˆ ⊗O ∞ ) ∼ = K top ∗ ( A nθ ). Now use Equation (11). (cid:3) We just determined the isomorphism type of the algebraic K-theory groups of A nθ ˆ ⊗O ∞ . Onecan also describe the elements in these groups using Rieffel’s results in [51]. Remark 5.3.
It follows from [51] that for irrational θ the projections in A nθ generate all ofK ( A nθ ˆ ⊗O ∞ ) andK ( A nθ ˆ ⊗O ∞ ) ∼ = K top1 ( A nθ ˆ ⊗O ∞ ) ∼ = K top1 ( A nθ ) ∼ ← U A nθ /U A nθ . Here
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