The coarse Pimsner-Voiculescu sequence

TL;DR

This paper derives a K-theory sequence for $C^{*}$-algebras with $bZ$-actions using equivariant coarse K-homology, and explores its extension to broader equivariant coarse homology theories.

Contribution

It introduces a new derivation of the Pimsner-Voiculescu sequence via equivariant coarse K-homology and investigates its applicability to general equivariant coarse homology theories.

Findings

Derived the Pimsner-Voiculescu sequence using equivariant coarse K-homology.

Explored the extension of this sequence to more general equivariant coarse homology theories.

Provided insights into the relationship between $C^{*}$-algebra K-theory and coarse geometry.

Abstract

We derive the Pimsner-Voiculescu sequence calculating the K-theory of a $C^{*}$- algebra with $\mathbb{Z}$-action using constructions with equivariant coarse K-homology theory. We then investigate to which extend this idea extends to more general equivariant coarse homology theories.