Wallman-type compactifications and function lattices

Let F be a vector lattice of bounded continuous real functions on X separating points from closed sets of X. The compactification e_F(X) obtained by embedding X in a real cube via the diagonal map is different, in general, from the Wallman compactification w(Z(F)). In this paper, it is shown that there exists a lattice Fz containing F such that w(Z(F))=w(Z(Fz))=e_Fz(X). In particular this implies that w(Z(F)) is greater than e_F(X). Conditions in order to obtain w(Z(F))=e_F(X) are given. Finally, we prove that, if aX is a compactification of X, such that Cl_aX(aX\X) is 0-dimensional, then there is an algebra A of bounded continuous real functions such that w(Z(A))= e_A(X)= aX.