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

Abstract

We develop an algebraic formalism for topological $\mathbb{T}$-duality. More precisely, we show that topological $\mathbb{T}$-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 $E_{\infty }$-operad starting from any strongly self-absorbing $C^{\ast }$-algebra $\mathcal{D}$. Then we show that there is a functorial topological K-theory symmetric spectrum construction ${\mathbf{K}}_{\Sigma }^{top}(-)$ on the category of separable $C^{\ast }$-algebras, such that ${\mathbf{K}}_{\Sigma }^{top}(\mathcal{D})$ is an algebra over this operad; moreover, ${\mathbf{K}}_{\Sigma }^{top}(A\hat{\otimes }\mathcal{D})$ 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 $C^{\ast }$-algebras. We also show that ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-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 $C^{\ast }$-algebras coming from number theory and that of ${\mathcal{O}}_{\infty }$-stabilized noncommutative tori.

18 pages, Dedicated to Professor Marc A. Rieffel on his 75th birthday; v2 added a section on T-duality; v3 added a new section on strongly self-absorbing operads and reorganised the original material; v4 section 1 expanded and a few corrections in section 3; v5 journal reference, DOI, and a bibitem added

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Cited by in corpus (3)

• Symmetric monoidal noncommutative spectra, strongly self-absorbing $C^{\ast }$-algebras, and bivariant homology
• $C^{\ast }$-algebraic drawings of dendroidal sets
• Algebraic K-theory and properly infinite C*-algebras