Functoriality of Rieffel's Generalised Fixed-Point Algebras for Proper Actions

TL;DR

This paper demonstrates that Rieffel's fixed-point algebra construction for proper actions can be formulated as functors between categories of C*-algebras, establishing natural isomorphisms with crossed-product functors and impacting non-abelian duality theory.

Contribution

It introduces functorial formulations of Rieffel's fixed-point algebra construction and shows their natural isomorphism with crossed-product functors, linking proper actions and Morita equivalences.

Findings

Rieffel's fixed-point algebra construction can be made into functors.

These functors are naturally isomorphic to crossed-product functors.

Applications to non-abelian duality for crossed products are demonstrated.

Abstract

We consider two categories of C*-algebras; in the first, the isomorphisms are ordinary isomorphisms, and in the second, the isomorphisms are Morita equivalences. We show how these two categories, and categories of dynamical systems based on them, crop up in a variety of C*-algebraic contexts. We show that Rieffel's construction of a fixed-point algebra for a proper action can be made into functors defined on these categories, and that his Morita equivalence then gives a natural isomorphism between these functors and crossed-product functors. These results have interesting applications to non-abelian duality for crossed products.