Naturality and Induced Representations

TL;DR

This paper explores the naturality of induction of covariant representations in C*-dynamical systems, framing it as a natural transformation between crossed-product functors and extending classical theorems.

Contribution

It formalizes the naturality of induction in C*-dynamical systems using category theory and extends crossed product constructions to establish functorial relationships.

Findings

Induction of covariant representations is a natural transformation.

Green's Imprimitivity Theorem characterizes when induction is a natural equivalence.

Various special cases previously obtained are unified under this framework.

Abstract

We show that induction of covariant representations for C*-dynamical systems is natural in the sense that it gives a natural transformation between certain crossed-product functors. This involves setting up suitable categories of C*-algebras and dynamical systems, and extending the usual constructions of crossed products to define the appropriate functors. From this point of view, Green's Imprimitivity Theorem identifies the functors for which induction is a natural equivalence. Various spcecial cases of these results have previously been obtained on an ad hoc basis.