Symmetric monoidal noncommutative spectra, strongly self-absorbing $C^*$-algebras, and bivariant homology

Abstract

Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^*$-algebras $\mathtt{SC^*_\infty}$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{SC^*_\infty}$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^*$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{SC^*_\infty}$ and give universal characterizations thereof. We construct a stable $\infty$-categorical model for bivariant connective E-theory and compute the connective E-theory groups of $\mathcal{O}_\infty$-stable $C^*$-algebras. We also introduce and study the nonconnective version of Quillen's nonunital K'-theory in the framework of stable $\infty$-categories. This is done in order to promote our earlier result relating topological $\mathbb{T}$-duality to noncommutative motives to the $\infty$-categorical setup. Finally, we carry out some computations in the case of stable and $\mathcal{O}_\infty$-stable $C^*$-algebras.