Naturality of Symmetric Imprimitivity Theorems

TL;DR

This paper proves that Morita equivalences arising from symmetric imprimitivity theorems for commuting group actions on C*-algebras are natural, meaning they are compatible with homomorphisms and induction.

Contribution

It establishes the naturality of Morita equivalences in symmetric imprimitivity theorems for proper, saturated commuting group actions on C*-algebras.

Findings

Morita equivalence is compatible with homomorphisms.

Morita equivalence is compatible with induction processes.

Main result confirms naturality of the symmetric imprimitivity theorem.

Abstract

The first imprimitivity theorems identified the representations of groups or dynamical systems which are induced from representations of a subgroup. Symmetric imprimitivity theorems identify pairs of crossed products by different groups which are Morita equivalent, and hence have the same representation theory. Here we consider commuting actions of groups $H$ and $K$ on a $C^*$-algebra which are saturated and proper as defined by Rieffel in 1990. Our main result says that the resulting Morita equivalence of crossed products is natural in the sense that it is compatible with homomorphisms and induction processes.