Dilating covariant representations of a semigroup dynamical system arising from number theory
type: Item
instance of: scholarly article
title: Dilating covariant representations of a semigroup dynamical system arising from number theory
zbMATH Open document ID: 1347.22006
author: Jaspar Wiart
published in: Integral Equations and Operator Theory
publication date: 2016-03-09T00:00:00Z
review text: Inspired by a paper by \textit{J. Cuntz} et al. [Math. Ann. 355, No. 4, 1383--1423 (2013; Zbl 1273.22008)], who studied the \(C^\ast\)-algebra \(\mathfrak{I}(R)\) generated by the left-regular representation of the \(ax+ b\)-semigroup of a number ring \(R\) on \(\ell^2(R\rtimes R^\times)\), the author of the paper under review is able to prove that \(\mathfrak{I}(R)\) can be realized as the \(C^\ast\)-envelope of the isometric semicrossed product of a certain semigroup dynamical system \((\mathcal{A}_R,\alpha,R^\times)\). He also shows how to explicitly dilate any representation of \(\mathcal{A}_R \times^{\text{is}}_\alpha R^\times\) to a representation of \(\mathfrak{I}(R)\).
reviewed by: József Sándor
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11M35
Mathematics Subject Classification ID: 47L65
Mathematics Subject Classification ID: 46L40
zbMATH DE Number: 6552392
zbMATH Keywords: semicrossed product
zbMATH Keywords: dilation
zbMATH Keywords: dynamical system
zbMATH Keywords: \(C^\ast\)-envelope
MaRDI profile type: Publication
full work available at URL: s00020-015-2263-0
OpenAlex ID: W2234665215
cites work: Operator algebras and measure preserving automorphisms
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DOI: 10.1007/S00020-015-2263-0
description: scientific article; zbMATH DE number 6552392