MaPhySto
The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

MPS-RR 2004-32
December 2004

# A Random Matrix Approach to the Lack of Projections in $C^*_{\mathrm{red}}(\mathbb{F}_2)$

by: Uffe Haagerup, Hanne Schultz , Steen Thorbjørnsen

## Abstract

In 1982 Pimsner and Voiculescu computed the $K_0$- and $K_1$-groups of the reduced group $C^*$-algebra $C^*_{\mathrm{red}}(F_k)$ of the free group $F_k$ on $k$ generators and settled thereby a long standing conjecture: $C^*_{\mathrm{red}}(F_k)$ 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 $C^*_{\mathrm{red}}(F_k)$ 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 $\operatorname{Ext}(C^*_{\mathrm{red}}(F_k))$ is not a group for $k\ge 2$. 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.

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