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

Abstract

Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^{\ast }$-algebras $\mathtt{S}{\mathtt{C}}_{\infty }^{\ast }$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{S}{\mathtt{C}}_{\infty }^{\ast }$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^{\ast }$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{S}{\mathtt{C}}_{\infty }^{\ast }$ 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 $C^{\ast }$-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 $C^{\ast }$-algebras.

26 pages; v2 revised in accordance with arXiv:1412.8370, corrections in Sections 3 and 4; v3 minor changes, to appear in J. Noncommut. Geom

References in corpus (9)

• Strongly self-absorbing C*-algebras are Z-stable
• Motivic structures in non-commutative geometry
• Simplicial nerve of an A-infinity category
• Dwyer-Kan localization revisited
• Higher nonunital Quillen K'-theory, KK-dualities and applications to topological $\mathbb{T}$-dualities
• Model structure on projective systems of $C^{\ast }$-algebras and bivariant homology theories
• Colocalizations of noncommutative spectra and bootstrap categories
• Algebraic K-theory, K-regularity, and T-duality of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• Algebraic K-theory and properly infinite C*-algebras

Cited by in corpus (4)

• Model structure on projective systems of $C^{\ast }$-algebras and bivariant homology theories
• Colocalizations of noncommutative spectra and bootstrap categories
• Algebraic K-theory, K-regularity, and T-duality of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• $C^{\ast }$-algebraic drawings of dendroidal sets