Algebraic K-theory, K-regularity, and T-duality of $\mathcal{O}_\infty$-stable $C^*$-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 $E_\infty$-operad starting from any strongly self-absorbing $C^*$-algebra $\mathcal{D}$. Then we show that there is a functorial topological K-theory symmetric spectrum construction ${\bf K}_\Sigma^{top}(-)$ on the category of separable $C^*$-algebras, such that ${\bf K}_\Sigma^{top}(\mathcal{D})$ is an algebra over this operad; 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 $C^*$-algebras. We also show that $\mathcal{O}_\infty$-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 $\mathcal{O}_\infty$-stabilized noncommutative tori.