Hilbert \(C^{*}\)-modules and projective representations associated with multipliers
type: Item
instance of: scholarly article
title: Hilbert \(C^{*}\)-modules and projective representations associated with multipliers
zbMATH Open document ID: 1121.46044
author: Jaeseong Heo
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-04T00:00:00Z
review text: The Naimark--Sz.--Nagy characterization of positive definite functions on groups and Stinespring's decomposition for completely positive maps on a \(C^*\)-algebra are well-known representation theorems. The covariant extension of Stinespring's theorem was given by \textit{V.\,Paulsen} [Mich.\ Math.\ J.\ 29, 131--142 (1982; Zbl 0507.46060)], and a more general version for equivariant completely bounded maps was established by \textit{I.\,Raeburn}, \textit{A.\,M.\thinspace Sinclair} and \textit{D.\,P.\thinspace Williams} [Pac.\ J.\ Math.\ 139, No.\,1, 155--194 (1989; Zbl 0692.46056)]. Furthermore, there are dilations associated with completely multi-positive maps, see [\textit{J.\,Heo}, J.~Oper.\ Theory 41, No.\,1, 3--22 (1999; Zbl 0994.46019)]. The author of the paper under review considers a unified approach to such dilations using the Kolmogorov decomposition for positive definite kernels and investigates the dilations associated with projective \(\sigma\)-representations, which generalize the above dilations. For a strictly \(U\)-covariant and completely positive map from a \(C^*\)-algebra \(B\) into the algebra \(L_A(X)\) of adjointable module maps on a right Hilbert \(C^*\)-module \(X\), the author also constructs a representation of a right Hilbert \(A\)-module.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 46L08
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L99
zbMATH DE Number: 5149577
zbMATH Keywords: Hilbert \(C^*\)-module
zbMATH Keywords: projective \(\sigma\)-representation
zbMATH Keywords: \(\sigma\)-positive definite
zbMATH Keywords: minimal Kolmogorov decomposition
zbMATH Keywords: covariant completely positive linear map
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.08.072
OpenAlex ID: W1970937214
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DOI: 10.1016/J.JMAA.2006.08.072
description: scientific article; zbMATH DE number 5149577