On free entropy dimension of finite von Neumann algebras.
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 \(\delta(X_1, \dots, X_n)\) \(\leqslant 1\).
scientific article; zbMATH DE number 1890452
scientific article