On the relative Dixmier property for inclusions of \(C^*\)-algebras
type: Item
instance of: scholarly article
title: On the relative Dixmier property for inclusions of \(C^*\)-algebras
zbMATH Open document ID: 0953.46027
published in: Journal of Functional Analysis
publication date: 2000-05-29T00:00:00Z
review text: The relative Dixmier property for inclusions of \(C^\ast\)-algebras is investigated. An inclusion \(B\subset A\) of \(C^\ast\)-algebras with a conditional expectation \(E:A\to B\) is said to have the finite index if there exists \(c>0\) with \(E(x)>cx\) for all \(x\in A\). The relative Dixmier property of the inclusion \(B\subset A\) means that, for any \(x\in A\), the norm closed convex hull of the orbit of \(x\) with respect to the unitary group of \(B\), contains elements from the relative commutant of \(B\) in \(A\). In the \(C^\ast\)-context the Dixmier property for a single unital \(C^\ast\)-algebra \(B\) is defined as follows: For each \(x\in B\) the closed convex hull of the set \(\{uxu^\ast\mid u\in B \) unitary \(\}\) contains a multiple of the unit. It is shown that the Dixmier property of a unital simple algebra \(B\) admitting at least one tracial state implies already the relative Dixmier property of the inclusion \(B\subset A\). The proof is based on results on the weak relative Dixmier property for inclusions of von Neumann algebras and on relating the Dixmier property for \(C^\ast\)-algebras to the weak Dixmier property for von Neumann algebras arising naturally by the GNS representation. Also, the relative Dixmier property for finite von Neumann algebras is studied. As an application of the results obtained it is proved that if a simple \(C^\ast\)-algebra \(B\) has the Dixmier property then the crossed product of \(B\) by a discrete group \(G\) acting properly outer on it has the Dixmier property relative to \(B\). This gives a stronger version of Connes' local Rohlin's theorem for outer automorphisms on type \(\text{II}_1\) factors. Besides, the interplay between simplicity and the Dixmier property of \(C^\ast\)-algebras is analysed. The paper extends and simplify many results along this line.
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L35
zbMATH DE Number: 1423456
zbMATH Keywords: inclusions of \(C^\ast\)-algebras
zbMATH Keywords: Dixmier property
zbMATH Keywords: trace
zbMATH Keywords: inclusions of \(C^*\)-algebras
zbMATH Keywords: conditional expectation
zbMATH Keywords: GNS representation
zbMATH Keywords: crossed product
zbMATH Keywords: Connes' local Rohlin's theorem
zbMATH Keywords: outer automorphisms on type \(\text{II}_1\) factors
author: Sorin Popa
reviewed by: Jan Hamhalter
MaRDI profile type: Publication
full work available at URL: jfan.1999.3536
OpenAlex ID: W2089853741
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DOI: 10.1006/JFAN.1999.3536
description: scientific article; zbMATH DE number 1423456