The range of an operator in C(X) and its representing stochastic kernel (Q1073317)

Let X and Y be compact metric spaces without isolated points, \(T\in B(C(X),C(y))\) can be written as \(Tf(y)=\int f d\mu_ y\), \(y\in Y\). In this paper the author proves that if \(\omega\) (y\(\cdot Y)\neq 0\) for a point of continuity of \(\omega\) (,Y), then every neighborhood of y contains a perfect K such that \(\chi_ KT: C(X)\to C(K)\) is surjective and there is a complemented subspace E of C(X) such that \(\chi_ KT|_ E: E\to C(K)\) is an isomorphism onto C(K). This result may be considered as a refinement of a theorem of \textit{H. P. Rosenthal} [Isr. J. Math. 13(1972), 361-378 (1973; Zbl 0253.46048)].