Extensions of positive definite functions on amenable groups

TL;DR

This paper investigates conditions under which positive definite functions defined on subsets of amenable groups can be extended to the entire group, providing new theoretical insights and applications.

Contribution

It establishes a new extension theorem for positive definite functions on amenable groups based on combinatorial properties of Cayley graphs.

Findings

Extension of positive definite functions under specific Cayley graph conditions

Derivation of known extension results as special cases

Introduction of new applications for positive definite function extensions

Abstract

Let $S$ be a subset of a amenable group $G$ such that $e\in S$ and $S^{-1}=S$. The main result of the paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as a corollary. New applications are also presented.