An Equivariant Theory for the Bivariant Cuntz Semigroup

TL;DR

This paper develops an equivariant extension of the bivariant Cuntz semigroup for C*-algebras with compact group actions, exploring its properties, connections, and providing a concrete presentation.

Contribution

It introduces an equivariant Cuntz semigroup theory, aligning with recent proposals, and offers new insights into its structure and applications.

Findings

Established functoriality properties of the equivariant Cuntz semigroup

Connected the theory with crossed product constructions

Provided a concrete open projection description

Abstract

We provide an equivariant extension of the bivariant Cuntz semigroup introduced in previous work for the case of compact group actions over C*-algebras. Its functoriality properties are explored and some well-known classification results are retrieved. Connections with crossed products are investigated, and a concrete presentation of equivariant Cuntz homology is provided. The theory that is here developed can be used to define the equivariant Cuntz semigroup. We show that the object thus obtained coincides with the recently proposed one by Gardella and Santiago, and we complement their work by providing an open projection picture of it.