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 is a hyperfinite von Neumann algebra with a normal, tracial state and is a set of selfadjoint generators for . We calculate , the modified free entropy dimension of . Moreover, we show that depends only on and . Consequently, is independent of the choice of generators for . In the course of the argument we show that if is a set of selfadjoint generators for a von Neumann algebra with a normal, tracial state and has finite-dimensional approximants, then for any hyperfinite von Neumann subalgebra of Combined with a result by Voiculescu, this implies that if has a regular diffuse hyperfinite von Neumann subalgebra, then .

**1.**Bernd Carl and Irmtraud Stephani,*Entropy, compactness and the approximation of operators*, Cambridge Tracts in Mathematics, vol. 98, Cambridge University Press, Cambridge, 1990. MR**1098497****2.**Ken Dykema,*Free products of hyperfinite von Neumann algebras and free dimension*, Duke Math. J.**69**(1993), no. 1, 97–119. MR**1201693**, 10.1215/S0012-7094-93-06905-0**3.**Liming Ge,*Applications of free entropy to finite von Neumann algebras. II*, Ann. of Math. (2)**147**(1998), no. 1, 143–157. MR**1609522**, 10.2307/120985**4.**Ge, Liming and Shen, Junhao,*On free entropy dimension of finite von Neumann algebras*, Geometric and Functional Analysis, Vol. 12, (2002), 546-566.**5.**Raymond, Jean Saint,*Le volume des idéaux d'opérateurs classiques*, Studia Mathematica, vol. LXXX (1984), 63-75.**6.**Stefan, M.,*The indecomposability of free group factors over nonprime subfactors and abelian subalgebras*, preprint.**7.**Stanisław J. Szarek,*Metric entropy of homogeneous spaces*, Quantum probability (Gdańsk, 1997) Banach Center Publ., vol. 43, Polish Acad. Sci., Warsaw, 1998, pp. 395–410. MR**1649741****8.**D. V. Voiculescu, K. J. Dykema, and A. Nica,*Free random variables*, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR**1217253****9.**Dan Voiculescu,*The analogues of entropy and of Fisher’s information measure in free probability theory. II*, Invent. Math.**118**(1994), no. 3, 411–440. MR**1296352**, 10.1007/BF01231539**10.**D. Voiculescu,*The analogues of entropy and of Fisher’s information measure in free probability theory. III. The absence of Cartan subalgebras*, Geom. Funct. Anal.**6**(1996), no. 1, 172–199. MR**1371236**, 10.1007/BF02246772**11.**Dan Voiculescu,*A strengthened asymptotic freeness result for random matrices with applications to free entropy*, Internat. Math. Res. Notices**1**(1998), 41–63. MR**1601878**, 10.1155/S107379289800004X

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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:
https://doi.org/10.1090/S0002-9947-03-03286-0

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