Topological equivariant coarse K-homology

TL;DR

This paper develops equivariant coarse homology theories for $C^*$-categories with group actions, using Roe categories and homological functors, impacting the understanding of assembly maps in operator algebras.

Contribution

It introduces new equivariant coarse homology theories for $C^*$-categories and demonstrates their application to properties of assembly maps in $K$-theory.

Findings

Constructed equivariant coarse homology theories for $C^*$-categories.

Showed certain functors are CP-functors on the orbit category.

Proved implications for the injectivity of assembly maps.

Abstract

For a $C^{*}$-category with a strict $G$-action we construct examples of equivariant coarse homology theories. To this end we first introduce versions of Roe categories of objects in $C^{*}$-categories which are controlled over bornological coarse spaces, and then apply a homological functor. These equivariant coarse homology theories are then employed to verify that certain functors on the orbit category are CP-functors. This fact has consequences for the injectivity of assembly maps.