A Class of Subalgebras of C(X) and the Associated Compactness

It was established by Plank [4] that the Stone-Cech compactification βX of a Tychonoff space X can be realized as the set of all maximal ideals of an arbitrary subalgebra A(X)≡ A of C(X) that contains C(X), equipped with the hull kernel toplogy. With every such A, we have associated a subset of βX, which is an A-analogue of , the Hewitt real compactification of X. X is called A-compact if = X. We have shown that can be identified with the set of all homomorphisms on A onto R, with the topology in herited from that of the product space ; the analogous results giving algebraic descriptions of vX and βX as considered in the monograph of Gillman and Jerison [3] follow simply on putting A = C(X) and A =C(X) respectively. We have also proved that a subset Y of βX containing X is real conmpact if and only if Y equals to some . We have investigated in some details the problem when for two subalgebras A, B of C(X) containing C(X), do we have = ; in case A ⊂ B, we have shown that = if and only if every homomorphism on A onto R extends to a homomorphism on B onto R, a special case of this result giving us a characterisation of pseudo-compactness