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

2011/10/15 by Rachid El Harti, Harti, Rachid El, N. Christopher Phillips +3
Mathematics · #22D15 #22D25 #46K05 #46K10 #46L05 (Secondary) #46M40 (Primary) 22D10 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1110.3411

openalex publication_date 2011/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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