Independent resolutions for totally disconnected dynamical systems II: C*-algebraic case

TL;DR

This paper introduces a method to compute the K-theory of crossed products from totally disconnected dynamical systems using independent resolutions, providing criteria for their existence and illustrating with examples.

Contribution

It develops the concept of independent resolutions in the C*-algebraic setting and offers criteria for finite length resolutions, advancing the analysis of such dynamical systems.

Findings

K-theory can be computed via six-term exact sequences

Criteria for existence of finite length resolutions

Illustrative examples demonstrating the approach

Abstract

We develop the notion of independent resolutions for crossed products attached to totally disconnected dynamical systems. If such a crossed product admits an independent resolution of finite length, then its K-theory can be computed (at least in principle) by analysing the corresponding six-term exact sequences. Building on our previous paper on algebraic independent resolutions, we give a criterion for the existence of finite length independent resolutions. Moreover, we illustrate our ideas in various concrete examples.