A Lévy-Khinchin Formula for the Space of Infinite Dimensional Square Complex Matrices

Abstract

Using a generalised Bochner type representation for Olshanski spherical pairs, we establish a Lévy-Khinchin formula for the continuous functions of negative type on the space $V_{\infty }=M(\infty ,\mathbb{C})$ of infinite dimensional square complex matrices 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 $V_n=M(n,\mathbb{C})$, 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.

16 pages, published in Bulletin des sciences Mathématiques 2015