[Paper Review] Crossed products, the Mackey-Rieffel-Green machine and applications

Siegfried Echterhoff|arXiv (Cornell University)|Jun 25, 2010

Advanced Operator Algebra Research105 references18 citations

TL;DR

This paper provides an accessible introduction to the ideal structure and representation theory of crossed products arising from actions of locally compact groups on C*-algebras, focusing on the Mackey-Rieffel-Green machinery for induced representations. It explains key concepts and ideas—without full proofs—offering foundational insights for researchers in operator algebras and noncommutative geometry.

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.

Motivation & Objective

• To clarify the ideal structure of crossed product C*-algebras arising from group actions on C*-algebras.
• To explain the representation theory of crossed products, particularly the theory of induced representations via the Mackey-Rieffel-Green machine.
• To provide conceptual clarity and intuitive understanding of advanced tools in operator algebra, especially for researchers new to the field.
• To serve as a preparatory guide for deeper study, referencing Dana Williams' comprehensive book as a primary source for detailed proofs.

Proposed method

• Utilizes the framework of group actions on C*-algebras to construct crossed product C*-algebras as universal objects satisfying certain covariance relations.
• Applies the Mackey machine to analyze induced representations from subgroups, focusing on the role of covariant representations and imprimitivity systems.
• Emphasizes structural insights over complete proofs, highlighting key ideas such as the correspondence between ideals and invariant subsets.
• Relies on the theory of induced representations via covariant systems, linking representation theory of group C*-algebras to those of crossed products.
• Uses the Rieffel correspondence to relate ideals in crossed products to invariant ideals in the original C*-algebra.
• Draws on the theory of induced representations to establish a bridge between representations of the group and representations of the crossed product.

Experimental results

Research questions

• RQ1 How do the ideals in a crossed product C*-algebra relate to the invariant ideals in the underlying C*-algebra?
• RQ2 What is the role of covariant representations in constructing induced representations for crossed products?
• RQ3 How does the Mackey-Rieffel-Green machine generalize the classical induced representation theory to the noncommutative setting?
• RQ4 In what way do the structure of the group action and the properties of the C*-algebra influence the representation theory of the crossed product?
• RQ5 How can the induced representation construction be used to classify irreducible representations of crossed products?

Key findings

• The ideal structure of a crossed product is closely tied to the invariant ideals of the original C*-algebra under the group action.
• The Mackey-Rieffel-Green machine provides a systematic method for constructing representations of crossed products via induction from subgroups.
• Covariant representations of the group-action system correspond bijectively to nondegenerate representations of the crossed product.
• The Rieffel correspondence establishes a lattice isomorphism between invariant ideals in the C*-algebra and ideals in the crossed product.
• Induced representations from closed subgroups yield irreducible representations of the crossed product under suitable conditions.
• The theory allows for a unified treatment of induced representations in both group C*-algebras and crossed products, extending classical Mackey theory.