On regular operators extending (pseudo)metrics

Abstract

It is proved that for every stratifiable space $Y$ and a closed subset $X\subset Y$ there exists a regular (i.e. linear positive with unit norm) extension operator $T:C(X\times X)\to C(Y\times Y)$ preserving the class of (pseudo)metrics. This operator is continuous with respect to the pointwise as well as to the compact-open topologies on the linear lattices of continuous functions $C(X\t X)$ and $C(Y\t Y)$. If moreover the space Y is metrizable then the operator $T$ preserves the class of admissible metrics. The equivariant analog of the above statement is proved as well.

This paper was written in 1992 but never published, as its main result was later superseded by the paper (arXiv:1202.1381) which, however, employs completely different methods to achieve the same goal, so I decided to make this old unpublished paper available for researchers