On uniformly continuous surjections between function spaces

Abstract We consider uniformly continuous surjections between $C_p(X)$ and $C_p(Y)$ (resp., $C_p^*(X)$ and $C_p^*(Y$ )) and show that if X has some dimensional-like properties, then so does Y . In particular, we prove that if $T:C_p(X)\to C_p(Y)$ is a continuous linear surjection and $\dim X=0$ , then ${\dim Y=0}$ . This provides a positive answer to a question raised by Kawamura–Leiderman [11, Problem 3.1].