On functions of negative type on the Olshanski spherical pair $(SL(\infty),SU(\infty))$

TL;DR

This paper proves that all continuous functions of negative type, which are bi-invariant under $SU()}$, are bounded on the infinite-dimensional special linear group $SL()}$, using a generalized Bochner representation.

Contribution

It introduces a generalized Bochner type representation for Olshanski spherical pairs and applies it to establish boundedness of negative type functions.

Findings

All $SU()}$-biinvariant continuous negative type functions on $SL()}$ are bounded.

The generalized Bochner representation is effective in analyzing functions on infinite-dimensional groups.

Abstract

In this paper, using a generalized Bochner type representation for Olshanski spherical pairs, we prove the boundedness of every $SU(\infty)$-biinvariant continuous function of negative type on the infinite dimensional special linear group $SL(\infty)$.