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

U. Haagerup; S. Thorbjørnsen

doi:10.4171/dm/63

JournalsdmVol. 4pp. 341–450

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

U. Haagerup
S. Thorbjørnsen

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Abstract

In this paper we find asymptotic upper and lower bounds for the spectrum of random operators of the form

S^{*} S = (i = 1 \sum r a_{i} \otimes Y_{i}^{(n)})^{*} (i = 1 \sum r a_{i} \otimes Y_{i}^{(n)}),

where $a_{1}, \dots, a_{r}$ are elements of an exact $C^{*}$ -algebra and $Y_{1}^{(n)}, \dots, Y_{r}^{(n)}$ 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:

Cite this article

U. Haagerup, S. Thorbjørnsen, Random matrices and $K$ -theory for exact $C^{*}$ -algebras. Doc. Math. 4 (1999), pp. 341–450

DOI 10.4171/DM/63