Equivariant representable K-theory

TL;DR

This paper develops a framework connecting equivariant Kasparov groups with representable K-theory, using classifying spaces and sigma-C*-algebras, and explores conditions for vector bundles to generate this theory.

Contribution

It introduces a novel interpretation of equivariant Kasparov groups as representable K-theory and provides methods to compute these groups via classifying spaces and sigma-C*-algebras.

Findings

Equivariant Kasparov groups can be interpreted as representable K-theory groups.

The paper provides a computation method using classifying spaces and sigma-C*-algebras.

Conditions are identified under which equivariant vector bundles generate the K-theory.

Abstract

We interpret certain equivariant Kasparov groups as equivariant representable K-theory groups. We compute these groups via a classifying space and as K-theory groups of suitable sigma-C*-algebras. We also relate equivariant vector bundles to these sigma-C*-algebras and provide sufficient conditions for equivariant vector bundles to generate representable K-theory. Mostly we work in the generality of locally compact groupoids with Haar system.