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The fundamental group of the von Neumann algebra of a free group with infinitely many generators is \({\mathbb{R}{}}_ +\backslash{}\{ 0\}\). (English) Zbl 0791.46036

The author uses the noncommutative (quantum) probability approach of D. Voiculescu [Operator algebras, unitary representations and invariant theory, Prog. Math. 92, 45-60 (1990; Zbl 0744.46055) and Invent. Math. 104, No. 1, 201-220 (1991; Zbl 0736.60007)] to prove the following
Theorem: The fundamental group \({\mathcal F}({\mathcal L} (F_ \infty))\) of the type \(\text{II}_ 1\) factor \({\mathcal L}(F_ \infty)\) of a free (noncommutative) group with infinitely many generators is the group \(\mathbb{R}_ + \setminus\{0\}\).
This result implies that for any \(s\in [4,\infty)\) there exists a subfactor of \({\mathcal L}(F_ \infty)\) of index \(s\) and furthermore, that for any \(\lambda\in (0,1)\) there exists a type \(\text{III}_ \lambda\) factor having a core isomorphic to \({\mathcal L} (F_ \infty) \overline{\otimes} B(H)\).

MSC:

46L10 General theory of von Neumann algebras
46L37 Subfactors and their classification
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L40 Automorphisms of selfadjoint operator algebras
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