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.