  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 $\dim Y=0$ if
$\dim X=0$. This provides a positive answer to a question raised by
Kawamura-Leiderman \cite[Problem 3.1]{kl}.
