Multiplicatively range-preserving maps between Banach function algebras
Let \(X,Y\) be locally compact Hausdorff spaces and let \(A,B\) be two Banach function algebras on \(X,Y\) respectively. \(T:A\rightarrow B\) is said to be multiplicatively range preserving if \((TfTg)(Y)=(fg)(X)\) for all \(f,g \in A\). The study of the structure of such maps has received a lot of attention. The earliest results in this direction were obtained by \textit{L.\,Molnár} [Proc.\ Am.\ Math.\ Soc.\ 130, No.\,1, 111--120 (2002; Zbl 0983.47024)] who described such maps, in the context of compact spaces, as composition operators. Let \(c(A)\) denote the Choquet boundary of \(A\). If all the points of \(c(A),c(B)\) are strong boundary points (which is the case, for example, when \(A\) and \(B\) are both completely regular), then \(T\) has the form \(f\rightarrow h f\circ\phi\) on \(c(B)\), for an onto homeomorphism \(\phi:c(B)\rightarrow c(A)\) and a real-valued continuous map \(h\) on \(c(B)\) with \(h^2=1\).
scientific article; zbMATH DE number 5568995
scientific article