Crossed products and the Mackey-Rieffel-Green machine

TL;DR

This paper introduces the ideal structure and representation theory of crossed products of C*-algebras by locally compact group actions, focusing on the Mackey-Rieffel-Green theory of induced representations.

Contribution

It provides an accessible overview of the Mackey-Rieffel-Green theory and its role in understanding the structure of crossed product C*-algebras.

Findings

Explains the main ideas behind the Mackey-Rieffel-Green theory.

Highlights the importance of induced representations in the structure of crossed products.

Provides references for further detailed study.

Abstract

We give an introduction into the ideal structure and representation theory of crossed products by actions of locally compact groups on C*-algebras. In particular, we discuss the Mackey-Rieffel-Green theory of induced representations of crossed products and groups. Although we do not give complete proofs of all results, we try at least to explain the main ideas. For a more detailed exposition of many of the results presented here we refer to the beautiful recent book by Dana Williams.