Linear continuous surjections of $C_p$-spaces over compacta

Abstract

Let $X$ and $Y$ be compact Hausdorff spaces and suppose that there exists a linear continuous surjection $T:C_p(X)\to C_p(Y)$, where $C_p(X)$ denotes the space of all real-valued continuous functions on $X$ endowed with the pointwise convergence topology. We prove that $\dim X=0$ implies $\dim Y=0$. This generalizes a previous theorem \cite[Theorem 3.4]{LLP} for compact metrizable spaces. Also we point out that the function space $C_p(P)$ over the pseudo-arc $P$ admits no densely defined linear continuous operator $C_p(P)\to C_p([0,1])$ with a dense image.

15 pages