  In 1982 Pimsner and Voiculescu computed the K_0- and K_1-groups of the
reduced group C*-algebra C*_red(F_k) of the free group F_k on k generators and
settled thereby a long standing conjecture: C*_red(F_k) 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
C*_red(F_k) 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 Ext(C*_red(F_k)) 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.
