The free entropy dimension of hyperfinite von Neumann algebras

By Kenley Jung. 2001

62 citations recorded by the source corpus.

Suppose M is a hyperfinite von Neumann algebra with a normal, tracial state φ and {α 1 ,...,α n } is a set of selfadjoint generators for M. We calculate δ 0 (α 1 ,...,α n ), the modified free entropy dimension of {α 1 ,...,α n }. Moreover, we show that δ 0 (α 1 ,...,α n ) depends only on M and Φ. Consequently, δ 0 (α 1 ,...,α n ) is independent of the choice of generators for M. In the course of the argument we show that if {b 1 ,...,b n } is a set of selfadjoint generators for a von Neumann algebra R with a normal, tracial state and {b 1 ,...,b n } has finite-dimensional approximants, then δ 0 (N) < δ 0 (b 1 ,...,b n ) for any hyperfinite von Neumann subalgebra N of R. Combined with a result by Voiculescu, this implies that if R has a regular diffuse hyperfinite von Neumann subalgebra, then δ 0 (b 1 ,....,b n ) = 1.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c9d978748d98c57081c0453988eb1f4555b7e01e

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Citations in the corpus

Free entropy dimension in amalgamated free products (2006, methodology): “Then [17] implies (after some tedious calculations) that”
L 2-homology for von Neumann algebras (2003, background): “The corresponding estimate for δ0 is a consequence of hyperfinite monotonicity of [Jun01].”
Some estimates for non-microstates free entropy dimension with applications to q-semicircular families (2003, background): “This fact has the following amusing consequence, which is the non-microstates analog of the “hyperfinite monotonicity” of [5].”
Strongly 1-Bounded Von Neumann Algebras (2005, methodology): “Recall that cutoff constants for the operator norm were used in the definition of K� (F )a ndP� (F )a nd that� quantities K�,R (F )a ndP�,R (F ) were introduced in [ J1 ] where the microstates spaces have cutoff constants.”