Random matrices and KK-theory for exact CC^*-algebras

  • U. Haagerup

  • S. Thorbjørnsen

Random matrices and $K$-theory for exact $C^*$-algebras cover
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In this paper we find asymptotic upper and lower bounds for the spectrum of random operators of the form SS=(i=1raiYi(n))(i=1raiYi(n)),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 a1,,ara_1,\ldots,a_r are elements of an exact CC^*-algebra and Y1(n),,Yr(n)Y_1^{(n)},\ldots,Y_r^{(n)} are complex Gaussian random n×nn\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:

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U. Haagerup, S. Thorbjørnsen, Random matrices and KK-theory for exact CC^*-algebras. Doc. Math. 4 (1999), pp. 341–450

DOI 10.4171/DM/63