The equivariant Cuntz semigroup

TL;DR

This paper introduces an equivariant version of the Cuntz semigroup that accounts for group actions, establishing structural properties, functoriality, and applications to classification and freeness characterization.

Contribution

It develops the equivariant Cuntz semigroup, proves its key properties, and applies it to classify actions and characterize freeness in topological and operator algebra contexts.

Findings

Equivariant Cuntz semigroup is a semimodule over the representation semiring.

It is continuous, stable, and functorial, with isomorphic invariants under cocycle conjugacy.

Applications include characterizing freeness of group actions and classifying certain operator algebra actions.

Abstract

We introduce an equivariant version of the Cuntz semigroup, that takes an action of a compact group into account. The equivariant Cuntz semigroup is naturally a semimodule over the representation semiring of the given group. Moreover, this semimodule satisfies a number of additional structural properties. We show that the equivariant Cuntz semigroup, as a functor, is continuous and stable. Moreover, cocycle conjugate actions have isomorphic associated equivariant Cuntz semigroups. One of our main results is an analog of Julg's theorem: the equivariant Cuntz semigroup is canonically isomorphic to the Cuntz semigroup of the crossed product. We compute the induced semimodule structure on the crossed product, which in the abelian case is given by the dual action. As an application of our results, we show that freeness of a compact Lie group action on a compact Hausdorff space can be characterized in terms of a canonically defined map into the equivariant Cuntz semigroup, extending results of Atiyah and Segal for equivariant $K$-theory. Finally, we use the equivariant Cuntz semigroup to classify locally representable actions on direct limits of one-dimensional NCCW-complexes, generalizing work of Handelman and Rossmann.