Continuous analogues of Fock space. II. The spectral \(C^*\)-algebra
type: Item
instance of: scholarly article
title: Continuous analogues of Fock space. II. The spectral \(C^*\)-algebra
zbMATH Open document ID: 0715.46039
DOI: 10.1016/0022-1236(90)90082-V
author: William Arveson
published in: Journal of Functional Analysis
publication date: 1990-01-01T00:00:00Z
review text: This is the second part of a study of a semigroup \(\{\alpha_ t\); \(t\geq 0\}\) of normal *-endomorphisms of the algebra B(H) of all bounded operators on a Hilbert space H such that \(\alpha_ t(1)=1\) for all \(t\geq 0\). In the first part [Mem. Am. Math. Soc. 80, No.409 (1989; Zbl 0697.46035)], the problem of classifying such semigroups up to outer conjugacy was reduced to the problem of classifying certain structure associated with them, called product systems. In this second part, the spectral \(C^*\) algebra associated with a product structure E is introduced as its topological spectral invariant. The author first introduces (Definition 2.12) a separable \(C^*\)-algebra \(C^*(E)\) whose non-degenerate *-representations corresponds to semigroups whose associated product systems are isomorphics to E. (Theorem 2.16 and Corollary 2 to Theorem 3.4). This \(C^*\)-algebra is nuclear (Corollary to Theorem 4.1). It is a continuous time analogue of the Cuntz algebra \({\mathcal O}_{\infty}.\) The author then defines its regular representations and, as its image there, the reduced \(C^*\)-algebra \(C^*_ r(E)\). It is shown that \(C^*_ r(E)\) have a nontrivial gauge-invariant closed two-sided ideals.
Mathematics Subject Classification ID: 46L99
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 4177905
zbMATH Keywords: continuous tensor product
zbMATH Keywords: semigroup of normal *-endomorphisms of the algebra B(H)
zbMATH Keywords: outer conjugacy
zbMATH Keywords: product systems
zbMATH Keywords: spectral \(C^ *\) algebra associated with a product structure
zbMATH Keywords: separable \(C^ *\)-algebra
zbMATH Keywords: continuous time analogue of the Cuntz algebra
zbMATH Keywords: reduced \(C^ *\)-algebra
zbMATH Keywords: gauge-invariant closed two-sided ideals
reviewed by: Huzihiro Araki
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4177905