Positive Linear Operators Continuous for Strict Topologies
J. A. Crenshaw · Proceedings of the American Mathematical Society · 1974
If $A$ is an $SW$ algebra of real-valued functions on a set $X$ equipped with the weak topology for $A$, and if $A$ separates its zero sets, then many results valued for ${C^b}(X)$ equipped with a strict topology remain true when $A$ is equipped with a strict topology. The concepts of $\alpha$-additivity and tight positive linear operators are introduced. It is shown that if $T$ is a positive linear map on $A$ into $z$-separating $SW$ algebra $B$ and if $T({1_A}) = {1_B}$, then there exists a continuous function $\phi$ on $Y$ (the domain of elements in $B$) into $X$ such that $Tf(y) = f(\phi (y))$ if and only if $T$ is an algebraic homomorphism and $\tau$-additive.