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 \mathtt E -theory and compute the connective \mathtt E -theory groups of \mathcal{O}_\infty -stable C^* -algebras. We also introduce and study the nonconnective version of Quillen's nonunital \mathtt 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.
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