On Linearly or Uniformly Continuous Surjections Between Cp-Spaces

Suppose there exists a linear continuous surjection T  from Cp(X) to Cp(Y) (resp., C^*p (X) to  C^*p (Y)), where Cp(X) (resp., C^*p (X)) denotes the space of all real-valued continuous (resp., continuous and bounded) functions on X endowed with the pointwise convergence topology.

We show that if X has some dimensional-like property P, then Y also has the same property. This is true if P is one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. Similar results are true if the surjection is uniformly continuous and inversely bounded.