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

摘要

We develop an algebraic formalism for topological $\mathbb{T}$-duality. Moreprecisely, we show that topological $\mathbb{T}$-duality actually induces anisomorphism between noncommutative motives that in turn implements thewell-known isomorphism between twisted K-theories (up to a shift). In order toestablish this result we model topological K-theory by algebraic K-theory. Wealso construct an $E_\infty$-operad starting from any strongly self-absorbing$C^*$-algebra $\mathcal{D}$. Then we show that there is a functorialtopological K-theory symmetric spectrum construction ${\bf K}_\Sigma^{top}(-)$on the category of separable $C^*$-algebras, such that ${\bfK}_\Sigma^{top}(\mathcal{D})$ is an algebra over this operad; moreover, ${\bfK}_\Sigma^{top}(A\hat{\otimes}\mathcal{D})$ 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 $C^*$-algebras. We also show that$\mathcal{O}_\infty$-stable $C^*$-algebras are K-regular providing evidence fora conjecture of Rosenberg. We conclude with an explicit description of thealgebraic K-theory of $ax+b$-semigroup $C^*$-algebras coming from number theoryand that of $\mathcal{O}_\infty$-stabilized noncommutative tori.