Joint work with V. Valov

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, then $\dim Y=0$ provided $\dim X=0$. This provides a partial answer to a question raised by Kawamura-Leiderman.