On regular operators extending (pseudo)metrics
Taras Banakh
TL;DR
The paper proves that for a stratifiable (and metrizable) space $Y$ with a closed subset $X$, there exists a regular extension operator $T:{\mathbb{R}}^{X\times X}\to {\mathbb{R}}^{Y\times Y}$ that preserves the class of bounded functions, continuous functions, and (pseudo)metrics, and that is tricontinuous with respect to relevant topologies. The core method introduces the squeezed join construction $SJ(X)$ and its infinite iterates $SJ^\infty(X)$, along with regular extension operators $sj$ and $sj^\infty$ that extend metrics while maintaining dominance and completeness properties. The main result is complemented by an invariant (equivariant) version: for a compact group $G$ acting on $Y$ with invariant $X$, there exists a regular extension operator on invariant functions obtained by averaging over $G$, i.e., $T(p)(y,y')=\int_G E(p)(gy,gy')\,d\mu$, preserving invariant metrics and their admissibility. Together, these results generalize Dugundji-type metric extensions to broad stratifiable settings and provide constructive, topology-preserving tools for extending metrics and invariant 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.