A noncommutative model for higher twisted K-Theory

TL;DR

This paper introduces a new operator algebraic model for twisted K-theory using strongly self-absorbing C*-algebras, encompassing all general twistings and connecting with homotopy theoretic descriptions.

Contribution

It develops a comprehensive noncommutative operator algebraic framework for twisted K-theory that includes all twistings classified by the unit spectrum, bridging with existing homotopy models.

Findings

Model includes all twistings classified by $bgl_1(KU)$

Establishes comparison with homotopy theoretic descriptions

Extends to twisted localizations like $KU[1/n]$ and $KU_{\mathbb{Q}}$

Abstract

We develop a operator algebraic model for twisted $K$-theory, which includes the most general twistings as a generalized cohomology theory (i.e. all those classified by the unit spectrum $bgl_1(KU)$). Our model is based on strongly self-absorbing $C^*$-algebras. We compare it with the known homotopy theoretic descriptions in the literature, which either use parametrized stable homotopy theory or $\infty$-categories. We derive a similar comparison of analytic twisted $K$-homology with its topological counterpart based on generalized Thom spectra. Our model also works for twisted versions of localizations of the $K$-theory spectrum, like $KU[1/n]$ or $KU_{\mathbb{Q}}$.