The Fundamental Group of the Von Neumann Algebra of a Free Group with Infinitely

In this paper we show that the fundamental group !T of the von Neumann algebra 2(Foo) of a free (noncommutative) group with infinitely many generators is lR+ \ {O}. This extends the result of Voiculescu who previously proved [26,27] that Q+ \ {O} is contained in !T(2(Foo)). This solves a classical problem in the harmonic analysis of the free group F 00. In particular, it follows that there exists subfactors of 2(Foo) with index s for every s E [4, 00). We will use the noncommutative (quantum) probabilistic approach introduced in Voiculescu's paper. Von Neumann algebras were introduced by Murray and von Neumann in the early thirties to provide a framework for quantum physics. As formulated by Heisenberg, quantization amounts to replacing the algebra of "observable" functions on the phase space of a classical system by a noncommuting algebra of infinite matrices, or more precisely, operators on a Hilbert space. By definition, a von Neumann algebra is a weakly closed, selfadjoint algebra of bounded operators on a Hilbert space that contains the identity operator. Where as any commutative von Neumann algebra is *-isomorphic to the algebra of bounded functions on a measure space, the structure of the noncommutative algebras is much more subtle. The simplest noncommutative von Neumann algebras are the n x n matrix algebras and the "hyperfinite algebras," i.e., the inductive limits of matrix algebras. Von Neumann algebras have provided a very powerful tool for studying noncommutative ergodic theory. More recently, Vaughan Jones has shown [13, 14] that the study of inclusions of von Neumann algebras naturally leads to new polynomial invariants for knot theory and to solutions of the Yang Baxter equation. Murray and von Neumann began by distinguishing three "type" classes for von Neumann algebras. Restricting to factors (von Neumann algebras with trivial center) on a separable Hilbert space, the type In factors have a dimension function on the projections (i.e., selfadjoint idempotents) which assumes the values {O, 1 , ... , n} (n ~ 00). Type II 1 (resp., 1100 ) algebras also have a dimension function, which assumes the values in [0,1] (resp., [0,00]). Such