Operator Algebras, Free Groups and Other Groups

The operator algebras associated to non commutative free groups have received a lot of attention, by F.J. Murray and J. von Neumann and by later workers. We review some properties of these algebras, both for free groups and for other groups such as lattices in Lie groups and Gromov hyperbolic groups. Our guideline is the following list of results for the free group Fn over n ≥ 2 generators. (1) W ∗ λ (Fn) is a full factor (Murray and von Neumann, 1943, rephrased by Connes). (2) There exists a continuous interpolating family (L(Fr))1<r≤∞ which satisfies L(Fn) = W ∗ λ (Fn) and the free product relation L(Fr) ⋆ L(Fr′) = L(Fr+r′) (Voiculescu, Dykema, Radulescu, 1990’s). (3) C λ (Fn) is a simple C-algebra with a unique trace (Powers, 1975). (4) C λ (Fn) is not nuclear (Takesaki, 1964). (5) C(Fn) is not exact (S. Wassermann, 1976). (6) C[Fn] has no zero divisor (G. Higman, 1940). (7) C λ (Fn) has no non trivial idempotent (Pimsner and Voiculescu, 1982). (8) The norm in C λ (Fn) of the sum of the generators of Fn and of their inverses is 1 n √ 2n− 1 (Kesten, 1959). (9) C λ (Fn) contains H(Fn)(Haagerup, 1979, rephrased by Jolissaint). We have collected a list of open problems; most of them are standard.