On uniformly continuous surjections between function spaces

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)→ C_p(Y) is a continuous linear surjection, then Y=0 if X=0. This provides a positive answer to a question raised by Kawamura-Leiderman .