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

Rachid El Harti; N. Christopher Phillips; Paulo R. Pinto

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

Operator Algebras 2012-04-23 v2

Authors: Rachid El Harti , N. Christopher Phillips , Paulo R. Pinto

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.

Keywords

profinite completion C*-algebra pro-C*-algebra profinite group unitary representation continuous representation

Cite

@article{arxiv.1110.3411,
  title  = {Profinite pro-C*-algebras and pro-C*-algebras of profinite groups},
  author = {Rachid El Harti and N. Christopher Phillips and Paulo R. Pinto},
  journal= {arXiv preprint arXiv:1110.3411},
  year   = {2012}
}

Comments

18 pages; no figures. Changes from the previous version: Questions 3.14 and 3.15 in the previous version have been answered. In this version, see Corollary 3.14, Proposition 3.16, and Examples 3.17 and 3.18. Some related changes have been made elsewhere. Some local reorganization has been done, and some misprints corrected. Houston J. Math., to appear