A random matrix approach to the lack of projections in

In 1982 Pimsner and Voiculescu computed the K0- and K1-groups of the reduced group C*-algebra of the free group Fk on k generators and settled thereby a long standing conjecture: has no projections except for the trivial projections 0 and 1. Later simpler proofs of this conjecture were found by methods from K-theory or from non-commutative differential geometry. In this paper we provide a new proof of the fact that is projectionless. The new proof is based on random matrices and is obtained by a refinement of the methods recently used by the first and the third named author to show that the semigroup is not a group for k⩾2. By the same type of methods we also obtain that two phenomena proved by Bai and Silverstein for certain classes of random matrices: “no eigenvalues outside (a small neighbourhood of) the support of the limiting distribution” and “exact separation of eigenvalues by gaps in the limiting distribution” also hold for arbitrary non-commutative selfadjoint polynomials of independent GUE, GOE or GSE random matrices with matrix coefficients.