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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\delta_0(b_1,\ldots,b_n)=1$.


References [Enhancements On Off] (What's this?)

  • 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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L54, 52C17, 53C30

Retrieve articles in all journals with MSC (2000): 46L54, 52C17, 53C30


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
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