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

TL;DR

This paper derives a Lévy-Khinchin formula for functions of negative type on the space of infinite-dimensional complex matrices, extending classical results to an infinite-dimensional setting using spherical pair representations.

Contribution

It introduces a Lévy-Khinchin formula for infinite-dimensional matrix spaces via a generalized Bochner representation, expanding the scope of harmonic analysis in infinite dimensions.

Findings

Established a Lévy-Khinchin formula for infinite-dimensional matrices

Extended classical harmonic analysis results to infinite-dimensional spaces

Utilized generalized spherical pair representations for the derivation

Abstract

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 $V_\infty= M(\infty, \mathbb C)$ 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.