On free entropy dimension of finite von Neumann algebras.
type: Item
instance of: scholarly article
title: On free entropy dimension of finite von Neumann algebras.
zbMATH Open document ID: 1040.46043
author: Junhao Shen
author: Li Ming Ge
published in: Geometric and Functional Analysis. GAFA
publication date: 2002-01-01T00:00:00Z
review text: Voiculescu has proved that some property \(T\) factors, including those arising from \(\text{SL}_n({\mathbb Z})\) \((n \geq 3)\), have generators with free entropy dimension not exceeding 1, cf. [J. Reine Angew. Math. 514, 113--118 (1999; Zbl 0959.46047)]. A natural question to ask is whether there are generators in those same property \(T\) factors whose free entropy dimension is greater than 1. The authors answer this question in this paper by showing that every set of generators of the group von Neumann algebras associated with \(\text{SL}_n({\mathbb Z})\) \(( n \geq 3)\) has free entropy dimension less than or equal to one. The same result is true for a large class of factors of type \(\text{II}_1\). The main theorem asserts that the free entropy dimension of any generators of a finite von Neumann algebra is less than or equal to one if there is at least one set of generators that satisfies a cyclic normalizing relation. The result generalizes the result stated in the authors' previous paper [Proc. Natl. Acad. Sci. USA 97, No. 18, 9881--9885 (2000; Zbl 0960.46042)]. More precisely, the main theorem says: Theorem. Let \((M, \tau)\) be a \(W^*\)-probability space. Assume that \(M\) is generated by a sequence of Haar unitary elements \(\{ U_j \}_j\). Suppose that \(N_j\) is the von Neumann subalgebra of \(M\) generated by \(\{ U_1, \dots, U_j \}\) and, for any \(j\), there is a Haar unitary element \(U_j'\) in \(N_j\) such that \(U_{j+1} U_j' U_{j+1}^*\) \(\in\) \(N_j\). Suppose that \(X_1, \dots, X_n\) are self-adjoint elements in \(M\) such that they generate \(M\) as a von Neumann algebra. Then the free entropy dimension \(\delta(X_1, \dots, X_n)\) \(\leqslant 1\).
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 94A17
zbMATH DE Number: 1890452
zbMATH Keywords: free entropy dimension
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: property T
zbMATH Keywords: generators
MaRDI profile type: Publication
full work available at URL: s00039-002-8256-6
OpenAlex ID: W1983037773
DOI: 10.1007/S00039-002-8256-6
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description: scientific article; zbMATH DE number 1890452