Are all linear continuous surjections between Cp-spaces c-good?

Determine whether every linear continuous surjection T: C_p(X) → C_p(Y), where X and Y are Tychonoff spaces and C_p(·) denotes the space of real-valued continuous functions endowed with the pointwise topology, is c-good for some constant c > 0; that is, ascertain whether there exists c > 0 such that for every bounded function g ∈ C(Y) there exists a bounded function f ∈ C(X) with T(f) = g and ||f|| ≤ c ||g|| (supremum norm).

Background

The paper develops dimension-theoretic consequences for uniformly continuous surjections between function spaces, with a central technical role played by c-good maps. A map is c-good if it admits uniform control of preimages of bounded functions in the supremum norm. This property is pivotal in deriving that various dimensional-like properties are preserved under such surjections.

While the authors prove that every linear continuous surjection between C_p^*(X) and C_p^*(Y) is c-good (via a Banach-space argument) and obtain several results assuming c-goodness, they explicitly state that it is unknown whether every linear continuous surjection T: C_p(X) → C_p(Y) must be c-good. They then give a partial result (Theorem 1.5) establishing that dim Y = 0 whenever dim X = 0 without assuming c-goodness, thereby addressing a specific instance of the broader unknown. The general c-goodness question for C_p remains open in this work.