Random Matrices and K-Theory for Exact $C^*$-Algebras

In this paper we find asymptotic upper and lower bounds for the spectrum of random operators of the form $$ S^*S=\Big(\sum_{i=1}^ra_i\otimes Y_i^{(n)}\Big)^* \Big(\sum_{i=1}^ra_i\otimes Y_i^{(n)}\Big), $$ where $a_1,\ldots,a_r$ are elements of an exact $C^*$-algebra and $Y_1^{(n)},\ldots,Y_r^{(n)}$ are complex Gaussian random $n\times n$ matrices, with independent entries. Our result can be considered as a generalization of results of Geman (1981) and Silverstein (1985) on the asymptotic behavior of the largest and smallest eigenvalue of a random matrix of Wishart type. The result is used to give new proofs of:

(1) Every stably finite exact unital $C^*$-algebra $\A$ has a tracial state.

(2) If $\A$ is an exact unital $C^*$-algebra, then every state on $K_0(\A)$ is given by a tracial state on $\A$.

The new proofs do not rely on quasitraces or on $AW^*$-algebra techniques.

1991 Mathematics Subject Classification: Primary 46L05; Secondary 46L50, 46L35, 46L80, 60F15.

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