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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The free entropy dimension of hyperfinite von Neumann algebras


Author: Kenley Jung
Journal: Trans. Amer. Math. Soc. 355 (2003), 5053-5089
MSC (2000): Primary 46L54; Secondary 52C17, 53C30
Published electronically: July 24, 2003
MathSciNet review: 1997595
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Abstract: Suppose $M$ is a hyperfinite von Neumann algebra with a normal, tracial state $\varphi$ and $\{a_1,\ldots,a_n\}$ is a set of selfadjoint generators for $M$. We calculate $\delta_0(a_1,\ldots,a_n)$, the modified free entropy dimension of $\{a_1,\ldots,a_n\}$. Moreover, we show that $\delta_0(a_1,\ldots,a_n)$ depends only on $M$ and $\varphi$. Consequently, $\delta_0(a_1,\ldots,a_n)$ is independent of the choice of generators for $M$. In the course of the argument we show that if $\{b_1,\ldots,b_n\}$ is a set of selfadjoint generators for a von Neumann algebra $\mathcal R$ with a normal, tracial state and $\{b_1,\ldots,b_n\}$has finite-dimensional approximants, then $\delta_0(N) \leq \delta_0(b_1,\ldots,b_n)$ for any hyperfinite von Neumann subalgebra $N$of $\mathcal R.$ Combined with a result by Voiculescu, this implies that if $\mathcal R$ has a regular diffuse hyperfinite von Neumann subalgebra, then $\delta_0(b_1,\ldots,b_n)=1$.


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Additional Information

Kenley Jung
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
Email: factor@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03286-0
PII: S 0002-9947(03)03286-0
Received by editor(s): March 4, 2002
Received by editor(s) in revised form: January 9, 2003
Published electronically: July 24, 2003
Additional Notes: Research supported in part by the NSF
Dedicated: For my parents
Article copyright: © Copyright 2003 American Mathematical Society