Diameter preserving mappings between function algebras
The authors consider function algebras \(A\subset C_0(X)\) satisfying that the extreme points of the unit ball of the quotient space \(A/C\), where \(C\) stands for the constant functions endowed with the diameter norm, are given by \(\alpha(\delta_{x}-\delta{x'})\) for some \(|\alpha|=1\) and \(x,x'\in Ch(A)\) (the Choquet boundary of \(A\)). They show that, for any diameter preserving map \(T:A\to B\) between function algebras \(A\subset C(X)\) and \(B\subset C(Y)\), with compact spaces \(X\) and \(Y\) such that \(A\) satisfies the above condition, there exist a linear functional \(L:A\to \mathbb C\), \(|\lambda|=1\), a set \(Y_0\subset Ch(B)\) and a continuous bijection \(\psi:Y_0\to Ch(A)\) such that \(Tf(y)= \lambda f(\psi(y))+L(f)\) for all \(y\in Y_0\) and \(f\in A\). Moreover, in the case when also \(B\) satisfies the above condition, then \(Y_0\) coincides with \(Ch(B)\) and the mapping \(\phi\) is an homomorphism. Results in the locally compact case are also provided and some applications are presented.
scientific article; zbMATH DE number 5984243