An excision theorem for the K-theory of C*-algebras, with applications to groupoid C*-algebras

TL;DR

This paper explores the relative K-theory of C*-algebras, establishing an excision theorem and applying it to groupoid C*-algebras, thereby advancing understanding of their algebraic and topological properties.

Contribution

It introduces an excision theorem for the K-theory of C*-algebras and applies it to groupoid C*-algebras, linking algebraic and topological structures.

Findings

Natural isomorphism between relative K-theories of related algebra pairs

Application of the theorem to groupoid C*-algebras with Haar systems

Enhanced tools for analyzing C*-algebra extensions

Abstract

We discuss the relative K-theory for a $C^{*}$-algebra, $A$, together with a $C^{*}$-subalgebra, $A' \subseteq A$. The relative group is denoted $K_{i}(A';A), i = 0, 1$, and is due to Karoubi. We present a situation of two pairs $A' \subseteq A$ and $B' \subseteq B$ are related so that there is a natural isomorphism between their respective relative K-theories. We also discuss applications to the case where $A$ and $B$ are $C^{*}$-algebras of a pair of locally compact, Hausdorff topological groupoids, with Haar systems.