Discrete product systems of Hilbert bimodules.
type: Item
instance of: scholarly article
title: Discrete product systems of Hilbert bimodules.
zbMATH Open document ID: 1059.46034
author: Neal J. Fowler
published in: Pacific Journal of Mathematics
publication date: 2002-10-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9904115
review text: Let \(A\) be a separable \(C^*\)-algebra. A Hilbert bimodule over \(A\) is a right Hilbert \(A\)-module \(X\) together with a \(*\)-homomorphism \(\varphi:A\to {\mathcal L}(X)\) which is used to define a left action of \(A\) on \(X\) via \(a.x:=\varphi(a)x\) for all \(a\in A\) and \(x\in X\). Here \({\mathcal L}(X)\) is the \(C^*\)-algebra of all adjointable operators on \(X\). Applying the idea of Arveson's continuous tensor product systems over \((0,\infty)\) [\textit{W. Arveson}, ``Continuous analogues of Fock space'' (Mem. Am. Math. Soc. 409) (1989; Zbl 0697.46035)], the author defines the notion of a product system of Hilbert bimodules and then a generalized Cuntz--Pimsner algebra \({\mathcal O}_X\). He associates with each twisted semigroup dynamical system \((A,P,\beta,\omega)\) a product system \(X=X(A,P,\beta,\omega)\) whose Cuntz--Pimsner algebra is isomorphic to the twisted semigroup crossed product \(A\rtimes_{\beta,\omega}P\). He also generalizes the concept of twisted crossed product. Among many interesting results, the author characterizes the faithful representations of \(B_P\rtimes_{\tau,X}P\) under specific assumptions by following the strategy used in Section 5 of \textit{N. J. Fowler} and \textit{I. Raeburn} [J. Funct. Anal. 155, 171--204 (1998; Zbl 0914.46054)].
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 46L08
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 1818996
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: Hilbert \(C^*\)-bimodule
zbMATH Keywords: product system
zbMATH Keywords: Cuntz--Pimsner algebra
zbMATH Keywords: twisted crossed product
zbMATH Keywords: Toeplitz extension
zbMATH Keywords: Toeplitz representation
MaRDI profile type: Publication
OpenAlex ID: W2044190082
arXiv ID: math/9904115
DOI: 10.2140/PJM.2002.204.335
description: scientific article