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Journal of Lie Theory 18 (2008), No. 3, 645--670
Copyright Heldermann Verlag 2008

Asymptotic Harmonic Analysis on the Space of Square Complex Matrices

Marouane Rabaoui
Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 175 rue de Chevaleret, 75013 Paris, France


This paper is largely of expository nature. We determine the spherical functions of positive type on the space $V_\infty= M(\infty, {\bf C})$ relatively to the action of the product group $K_\infty = U(\infty)\times U(\infty)$. The space $V_\infty$ is the inductive limit of the spaces of square complex matrices $V_n=M(n, {\bf C})$, and the group $K_\infty$ is the inductive limit of the product groups $K_n=U(n) \times U(n)$, where $U(n)$ is the unitary group.

Keywords: Square complex matrices, unitary group, inductive limit, function of positive type, spherical function, ergodic measure, generalized Bochner theorem.

MSC: 22E30; 43A35, 43A85, 43A90

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