Groups and Operator Algebras Seminar

Speaker:  Félix Parraud (Mittag-Leffler Institute and KTH)

Title:  The spectrum of tensor of random and deterministic matrices

Abstract:  In this talk, we consider operator-valued polynomials in Gaussian Unitary Ensemble random matrices.  I will explain a new strategy to bound its $L^p$-norm, up to an asymptotically small error, by the operator norm of the same polynomial evaluated in free semicircular variables as long as $p = o(N^{2/3})$. As a consequence, if the coefficients are $M$-dimensional matrices with $M = \exp(o(N^{2/3}))$, then the operator norm of this polynomial converges towards the one of its free counterpart. In particular this provides another proof of the Peterson-Thom conjecture thanks to the result of Ben Hayes.

The approach that we take in this paper is based on an asymptotic expansion obtained in a previous paper combined with a new result of independent interest on the norm of the composition of the multiplication operator and a permutation operator acting on a tensor of $\mathrm{C}^∗$-algebras.