On regular operators extending (pseudo)metrics

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 $C(X\text{t}X)$ and $C(Y\text{t}Y)$. 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.