A L\'evy-Khinchin Formula for the Space of Infinite Dimensional Square Complex Matrices

Using a generalised Bochner type representation for Olshanski spherical pairs, we establish a L'evy-Khinchin formula for the continuous functions of negative type on the space $V_{\mathrm{\infty }}=M(\mathrm{\infty },\mathbb{C})$ of infinite dimensional square complex matrices relatively to the action of the product group $K_{\mathrm{\infty }}=U(\mathrm{\infty })\times U(\mathrm{\infty })$. The space $V_{\mathrm{\infty }}$ is the inductive limit of the spaces $V_n=M(n,\mathbb{C})$, and the group $K_{\mathrm{\infty }}$ is the inductive limit of the product groups $K_n=U(n)\times U(n)$, where $U(n)$ is the unitary group.