Twisted K-theory, K-homology, and bivariant Chern–Connes type character of some infinite dimensional spaces

Mahanta, Snigdhayan

doi:10.1215/21562261-2693460

Cited by 5 publications

(5 citation statements)

References 63 publications

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“…). This KK-theory will agree with the bivariant K-theory for separable σ-C * -algebras [16] that was denoted by σ-kk-theory in [36] (not to be confused with the diffotopy invariant bivariant K-theory for locally convex algebras) on a reasonably large subcategory (cf. Theorem 5.9 below and Proposition 36 of [36]).…”

Section: 3supporting

confidence: 53%

Model structure on projective systems of $C^*$-algebras and bivariant homology theories

Barnea,

Joachim,

Mahanta

2015

Preprint

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Using the machinery of weak fibration categories due to Schlank and the first author, we construct a convenient model structure on the pro-category of separable C *algebras Pro(SC * ). The opposite of this model category models the ∞-category of pointed noncommutative spaces NS * defined by the third author. Our model structure on Pro(SC * ) extends the well-known category of fibrant objects structure on SC * . We show that the pro-category Pro(SC * ) also contains, as a full coreflective subcategory, the category of pro-C * -algebras that are cofiltered limits of separable C * -algebras. By stabilizing our model category we produce a general model categorical formalism for triangulated and bivariant homology theories of C * -algebras (or, more generally, that of pointed noncommutative spaces), whose stable ∞-categorical counterparts were constructed earlier by the third author. Finally, we use our model structure to develop a bivariant K-theory for all projective systems of separable C * -algebras generalizing the construction of Bonkat and show that our theory naturally agrees with that of Bonkat under some reasonable assumptions. Contents 36 A.1. Simplicial model categories 37 A.2. Left and right proper model categories 38 A.3. Pointed simplicial model categories 38 A.4. Left Bousfield localizations of model categories 39 A.5. Stabilization of model categories 40 Appendix B. ∞-categories 41 B.1. Relative categories and their associated ∞-categories 41 B.2. Stabilization of ∞-categories 42 References 45The first item codifies the requirement that one must build a homotopy theory for the prevalent notion in the literature. The second item takes into account the requirement that there should be minimal deviation from the well established theory of C * -algebras. The third item stipulates that the model category possess features that facilitate homotopy theoretic constructions therein (see Appendix A for more detail).We are aware of the following different model categories in the context of C * -algebras (the authors apologise for any omission due to ignorance):(1) The homotopy theory of cubical C * -spaces by Østvaer [45], (2) The model structure on ν-sequentially complete l.m.c-C * -algebras by Joachim-Johnson [30], (3) The model category (or the ∞-category) of pointed noncommutative spaces by the third author [38], (4) The Morita homotopy theory of C * -categories by Dell'Ambrogio-Tabuada [17], and (5) The operadic model structure on the topos P(SC * un op ) by the third author, where SC * un denotes the category of nonzero separable and unital C * -algebras with unital * -homomorphisms [35].

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Section: 3supporting

confidence: 53%

Model structure on projective systems of $C^*$-algebras and bivariant homology theories

Barnea,

Joachim,

Mahanta

2015

Preprint

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“…Example 3.15. In Example 11 of [39] it is explained how one can construct an inverse system of separable C * -algebras {C n } n∈N = {(CT(SU(n)), ι * n (P ))} n∈N starting from a principal P Ubundle P on SU(∞). The inverse limit of this diagram in topological * -algebras is the noncommutative twisted version of SU(∞).…”

Section: 2mentioning

confidence: 99%

Noncommutative stable homotopy and stable infinity categories

Mahanta

2014

J. Topol. Anal.

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The noncommutative stable homotopy category $\mathtt{NSH}$ is a triangulated category that is the universal receptacle for triangulated homology theories on separable $C^*$-algebras. We show that the triangulated category $\mathtt{NSH}$ is topological as defined by Schwede using the formalism of (stable) infinity categories. More precisely, we construct a stable presentable infinity category of noncommutative spectra and show that $\mathtt{NSH}^{op}$ sits inside its homotopy category as a full triangulated subcategory, from which the above result can be deduced. We also introduce a presentable infinity category of noncommutative pointed spaces that subsumes $C^*$-algebras and define the noncommutative stable (co)homotopy groups of such noncommutative spaces generalizing earlier definitions for separable $C^*$-algebras. The triangulated homotopy category of noncommutative spectra admits (co)products and satisfies Brown representability. These properties enable us to analyse neatly the behaviour of the noncommutative stable (co)homotopy groups with respect to certain (co)limits. Along the way we obtain infinity categorical models for some well-known bivariant homology theories like $\mathrm{KK}$-theory, $\mathrm{E}$-theory, and connective $\mathrm{E}$-theory via suitable (co)localizations. The stable infinity category of noncommutative spectra can also be used to produce new examples of generalized (co)homology theories for noncommutative spaces.Comment: 26 pages; v2 major revision with some improved results, a mistake in matrix homotopy computation removed, title slightly changed (extended); v3 discussion of semigroup C*-algebras removed and made a part of arXiv:1403.4130, Section 3 rewritten accordingly and the title shortened to reflect the changes; v4 final revision incorporating the referee's corrections; v5 added/updated appendix/reference

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“…More precisely, given any pair (E, h) with E locally compact one can construct a noncommutative stable 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 and showed that under favourable circumstances if B and B ′ are T-dual C * -algebras, then there is an invertible element in KK 0 (B, ΣB ′ ) that implements the twisted K-theory isomorphism (as in (2)).…”

Section: Our Resultsmentioning

confidence: 99%

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Mahanta

2015

Math Phys Anal Geom

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Abstract. We develop an algebraic formalism for topological T-duality. More precisely, we show that topological T-duality actually induces an isomorphism between noncommutative motives 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 algebraic K-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 spectrum construction K top Σ (−) on the category of separable C * -algebras, such that K top Σ (D) is an algebra over this operad; moreover, K top Σ (A⊗D) is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C * -algebras. We also show that O ∞ -stable C * -algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of ax + b-semigroup C * -algebras coming from number theory and that of O ∞ -stabilized noncommutative tori. IntroductionWithin the category of separable C * -algebras SC * the stable ones, i.e., those A ∈ SC * satisfying A⊗K ∼ = A, play a privileged role. For instance, it is known that they satisfy the Karoubi conjecture and appear very naturally in the context of twisted K-theory. The results in this article demonstrate that O ∞ -stable separable C * -algebras, i.e., those A ∈ SC * satisfying A⊗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 has promoted rich interaction between geometry, algebra, and (higher) category theory. Mirror symmetry is related to T-duality via the Strominger-Yau-Zaslow conjecture [61] and hence we 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 lot of attention in the mathematical literature. It is insensitive to subtle geometric structures but its mathematical underpinnings are very well understood [8,7,9]. One of the objectives of this article is to develop an algebraic formalism for topological T-duality relating it to the theory of noncommutative motives [34,35,65,45]. Along the way we obtain several interesting applications to algebraic K-theory and K-regularity of C * -algebras. The novelty 2010 Mathematics Subject Classification. 46L85, 19Dxx, 46L80, 57R56.

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Twisted K-theory, K-homology, and bivariant Chern–Connes type character of some infinite dimensional spaces

Cited by 5 publications

References 63 publications

Model structure on projective systems of $C^*$-algebras and bivariant homology theories

Model structure on projective systems of $C^*$-algebras and bivariant homology theories

Noncommutative stable homotopy and stable infinity categories

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

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