The analogues of entropy and of Fisher's information measure in free probability theory. III: The absence of Cartan subalgebras
type: Item
instance of: scholarly article
title: The analogues of entropy and of Fisher's information measure in free probability theory. III: The absence of Cartan subalgebras
zbMATH Open document ID: 0856.60012
DOI: 10.1007/BF02246772
published in: Geometric and Functional Analysis. GAFA
publication date: 1996-07-08T00:00:00Z
full work available at URL: https://eudml.org/doc/58223
review text: [For parts I and II see Commun. Math. Phys. 155, No. 1, 71-92 (1993; Zbl 0781.60006) and Invent. Math. 118, No. 3, 411-440 (1994; Zbl 0820.60001).] In part II we introduced the free entropy \(\chi(X_1,\dots, X_n)\) for an \(n\)-tuple of selfadjoint random variables in a tracial \(W^*\)-probability space. Here, we deal with certain technical questions about free entropy and with applications to some von Neumann algebra problems. It has been a longstanding open question whether every separable \(\text{II}_1\)-factor is generated by the normalizer of a maximal Abelian *-algebra (such algebras are called Cartan subalgebras). An equivalent question, which also provided the motivation, is, roughly stated, whether every separable \(\text{II}_1\)-factor arises from a measurable equivalence relation [see \textit{J. Feldman} and \textit{C. C. Moore}, Trans. Am. Math. Soc. 234, 289-324 (1977; Zbl 0369.22009) for the construction]. Using free entropy we provide a negative answer for the free group factors \({\mathcal L}(F_n)\) \((2\leq n\leq \infty)\). Actually, we prove a stronger result: the free group factors are not generated by the normalizer of any diffuse hyperfinite subalgebra.
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 94A17
zbMATH DE Number: 881250
zbMATH Keywords: free entropy
zbMATH Keywords: von Neumann algebra problems
zbMATH Keywords: free group factors
zbMATH Keywords: hyperfinite subalgebra
author: Dan Voiculescu
MaRDI profile type: Publication
cites work: Quelques propriétés des suites centrales dans les facteurs de type II\(_ 1\)
cites work: Ergodic Equivalence Relations, Cohomology, and Von Neumann Algebras. I
cites work: Q3671706
cites work: Limit laws for random matrices and free products
cites work: The analogues of entropy and of Fisher's information measure in free probability theory. II
cites work: Free Random Variables
OpenAlex ID: W2026938829
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description: scientific article; zbMATH DE number 881250