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

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 $E_{\mathrm{\infty }}$-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 ${\mathbf{K}}_{\mathrm{\Sigma }}^{top}(-)$ on the category of separable $C^{\ast }$-algebras, such that ${\bf K}\Sigma^{top}($\mathcal{D}$)$isanalgebraoverthisoperad;moreover,${\bf K}\Sigma^{top}(A\hat{\otimes}$\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 ${\mathcal{O}}_{\mathrm{\infty }}$-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 ${\mathcal{O}}_{\mathrm{\infty }}$-stabilized noncommutative tori.