Isomorphism of T"(X) to C(Y)

Determine the necessary and sufficient conditions on a topological space X under which the ring T"(X) is isomorphic to C(Y) for some topological space Y.

Background

The paper proves that T"(X) is generally not closed under uniform limits (e.g., for the real line and various separable normed linear spaces), which prevents it from being isomorphic to C(Y) in those cases.

The authors ask for a general characterization of when such an isomorphism exists.