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Applications of free entropy to finite von Neumann algebras. II. (English) Zbl 0924.46050

The old problem on the existence of prime Type \(II_1\) factor (with separable predual) is solved by using Voiculescu’s free probability theory. More precisely, it is shown that the Type \(II_1\) factor \({\mathcal L}_{F_n}\) resulting from the regular representation of the discrete free group \(F_n\) on \(n \) generators (\(n\geq 2\)) is not isomorphic to any tensor product of two factors of Type \(II_1\). (Such Type \(II_1\) factors are called prime). The proof relies on estimates of the free entropy for \(n\)-tuples of elements in a Type \(II_1\) factors and uses, among others, geometric methods based on the Grassmann manifolds. Estimates obtained enable to prove that the free entropy \(\chi(X_1,X_2,\ldots, X_n)\) where \(X_1,X_2,\ldots, X_n\) (\(n\geq 2)\) are self-adjoint elements generating a tensor product of two Type \(II_1\) factors equals to \(-\infty\). Since there are such elements in \({\mathcal L}_{F_n}\) with finite entropy the factors \({\mathcal L}_{F_n}\) are prime. It is also shown that the interpolated free group factors are prime. The results of the paper are considerably deep and open new perspectives in the theory of von Neumann factors.
[For part I see Am. J. Math. 119, No. 2, 467-485 (1997; Zbl 0871.46031)].

MSC:

46L55 Noncommutative dynamical systems
46L10 General theory of von Neumann algebras
46L35 Classifications of \(C^*\)-algebras

Citations:

Zbl 0871.46031
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