Profinite pro-C*-algebras and pro-C*-algebras of profinite groups

TL;DR

This paper introduces the concept of profinite completions for C*-algebras and groups, exploring their properties and the relationship between their representations, with some open questions remaining.

Contribution

It defines the pro-C*-algebra of a profinite group and establishes the correspondence between continuous representations and finite group factorizations.

Findings

Continuous representations correspond to finite group factorizations

Homomorphisms from group C*-algebras to pro-C*-algebras are characterized

Conditions for injectivity and surjectivity are discussed

Abstract

We define the profinite completion of a C*-algebra, which is a pro-C*-algebra, as well as the pro-C*-algebra of a profinite group. We show that the continuous representations of the pro-C*-algebra of a profinite group correspond to the unitary representations of the group which factor through a finite group. We define natural homomorphisms from the C*-algebra of a locally compact group and its profinite completion to the pro-C*-algebra of the profinite completion of the group. We give some conditions for injectivity or surjectivity of these homomorphisms, but an important question remains open.