zbMATH — the first resource for mathematics

The analogues of entropy and of Fisher’s information measure in free probability theory. III: The absence of Cartan subalgebras. (English) Zbl 0856.60012
[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 J. Feldman and 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.

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
94A17 Measures of information, entropy
Full Text: DOI EuDML
[1] J. Dixmier, Quelques proprietes des suites centrales dans les facteurs de type II, Inventiones Math. 7 (1969), 215–225. · Zbl 0174.18702
[2] J. Feldman, C.C. Moore, Ergodic equivalence relations, cohomology and von Neumann algebras, Trans. Amer. Math. Soc. 234 (1977), 289–361. · Zbl 0369.22009
[3] S. Popa, Orthogonal pairs of subalgebras in finite factors, J. Operator Theory 9 (1983), 253–268. · Zbl 0521.46048
[4] S.J. Szarek, Nests of Grassmann manifolds and orthogonal group, Proceedings of Research Workshop on Banach Space Theory (Bor-Luh-Lin, ed.), The University of Iowa, June 29–31 (1981), 169–185.
[5] D. Voiculescu, Limit laws for random matrices and free products, Inventiones Math. 104 (1991), 210–220. · Zbl 0736.60007
[6] D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory II, Inventiones Math. 118 (1994), 411–440. · Zbl 0820.60001
[7] D. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, CRM Monograph Series, vol. I American Math. Soc. 1992. · Zbl 0795.46049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.