Twisted K-theory with coefficients in C*-algebras

TL;DR

This paper develops a noncommutative generalization of twisted K-theory using Morita bundle gerbes, extending classical concepts to include nonabelian cohomology and higher twists, especially for the infinite Cuntz algebra.

Contribution

It introduces Morita bundle gerbes as a new twist in K-theory with C*-algebra coefficients, generalizing the classical bundle gerbe approach to noncommutative settings.

Findings

Defined Morita bundle gerbes and their stable equivalence classes.

Extended twisted K-theory descriptions to nonabelian cohomology.

Potential interpretation of higher twists for K-theory with the Cuntz algebra.

Abstract

We introduce a twisted version of $K$-theory with coefficients in a $C^*$-algebra $A$, where the twist is given by a new kind of gerbe, which we call Morita bundle gerbe. We use the description of twisted $K$-theory in the torsion case by bundle gerbe modules as a guideline for our noncommutative generalization. As it turns out, there is an analogue of the Dixmier-Douady class living in a nonabelian cohomology set and we give a description of the latter via stable equivalence classes of our gerbes. We also define the analogue of torsion elements inside this set and extend the description of twisted $K$-theory in terms of modules over these gerbes. In case $A$ is the infinite Cuntz algebra, this may lead to an interpretation of higher twists for $K$-theory.