Positive extensions of Schur multipliers

TL;DR

This paper studies how to extend partially defined positive Schur multipliers to fully defined ones and applies these results to extend positive definite functions on groups.

Contribution

It introduces conditions for extending partially defined Schur multipliers to positive ones and applies these to group positive definite function extensions.

Findings

Necessary and sufficient conditions for extensions of Schur multipliers.

Characterization of operator systems related to domains.

Application to extending positive definite functions on groups.

Abstract

We introduce partially defined Schur multipliers and obtain necessary and sufficient conditions for the existence of extensions to fully defined positive Schur multipliers, in terms of operator systems canonically associated with their domains. We use these results to study the problem of extending a positive definite function defined on a symmetric subset of a locally compact group to a positive definite function defined on the whole group.