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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.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
94A17 Measures of information, entropy
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