Assembly maps with coefficients in topological algebras and the integral K-theoretic Novikov conjecture

Mahanta, Snigdhayan

doi:10.1007/s40062-013-0027-6

Cited by 2 publications

(1 citation statement)

References 37 publications

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“…Remark 4.3. Using Proposition 2.2 and Theorem 4.1 the reduction principle for assembly maps (see Theorem 1.1 of [42]) can be generalized to include O ∞ as coefficients, i.e., for a countable, discrete, and torsion free group G, if the Baum-Connes assembly map with complex coefficients is injective (resp. split injective), then the Farrell-Jones assembly map in algebraic K-theory with O ∞ -coefficients is also injective (resp.…”

Section: K-regularity Of Omentioning

confidence: 99%

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

Mahanta

2013

Preprint

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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.

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Section: K-regularity Of Omentioning

confidence: 99%

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

Mahanta

2013

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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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Assembly maps with coefficients in topological algebras and the integral K-theoretic Novikov conjecture

Cited by 2 publications

References 37 publications

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

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

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

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