On linear operators extending [pseudo]metrics

TL;DR

This paper constructs positive linear extension operators for (pseudo)metrics from closed subsets of stratifiable or metrizable spaces to the entire space, preserving various properties and topologies, with an equivariant version also provided.

Contribution

It introduces a method for extending (pseudo)metrics via positive linear operators that preserve key properties and topologies, including an equivariant extension.

Findings

Extension operators preserve constant, bounded, and continuous functions.

Operators are continuous in multiple topologies: point-wise, uniform, and compact-open.

An equivariant extension operator is also established.

Abstract

For every closed subset $X$ of a stratifiable [resp. metrizable] space $Y$ we construct a positive linear extension operator $T:R^{X\times X}\to R^{Y\times Y}$ preserving constant functions, bounded functions, continuous functions, pseudometrics, metrics, [resp. dominating metrics, and admissible metrics]. This operator is continuous with respect to each of the three topologies: point-wise convergence, uniform, and compact-open. An equivariant analog of the above statement is proved as well.