Symmetric monoidal noncommutative spectra, strongly self-absorbing $C^{\ast }$-algebras, and bivariant homology

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