Rădulescu, Florin 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 J. Am. Math. Soc. 5, No. 3, 517-532 (1992). 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 followingTheorem: 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 \({\mathcal L} (F_ \infty) \overline{\otimes} B(H)\). Reviewer: M.Fritzsche (Potsdam) Cited in 13 Documents 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 Keywords:noncommutative (quantum) probability; type \(\text{II}_ 1\) factor; subfactor Citations:Zbl 0736.60007; Zbl 0744.46055 PDFBibTeX XMLCite \textit{F. Rădulescu}, J. Am. Math. Soc. 5, No. 3, 517--532 (1992; Zbl 0791.46036) Full Text: DOI