A Lévy-Khinchin formula for the space of infinite dimensional square complex matrices

Using a generalized Bochner type representation for Olshanski spherical pairs, we establish a Lévy-Khinchin formula for the continuous functions of negative type on the space V∞=M(∞,C) of infinite dimensional square complex matrices relatively to the action of the product group K∞=U(∞)×U(∞). The space V∞ is the inductive limit of the spaces Vn=M(n,C), and the group K∞ is the inductive limit of the product groups Kn=U(n)×U(n), where U(n) is the unitary group.