Random matrices and $K$-theory for exact $C^*$-algebras

Abstract

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: