REPRESENTATION THEOREMS FOR NORMED ALGEBRAS

We show that for a normal locally-P space X (where P is a topological property subject to some mild requirements) the subset C (P)(X) of C-b(X) consisting of those elements whose support has a neighborhood with P, is a subalgebra of C-b(X) isometrically isomorphic to C-c(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y is explicitly constructed as a subspace of the Stone-Cech compactification beta X of X and contains X as a dense subspace. Under certain conditions, C (P)(X) coincides with the set of those elements of C-b(X) whose support has P, it moreover becomes a Banach algebra, and simultaneously, Y satisfies C-c(Y) = C-0(Y). This includes the cases when P is the Lindelof property and X is either a locally compact paracompact space or a locally-P metrizable space. In either of the latter cases, if X is non-P, then Y is nonnormal and C (P) (X) fits properly between C-0(X) and C-b(X); even more, we can fit a chain of ideals of certain length between C-0(X) and C-b(X). The known construction of Y enables us to derive a few further properties of either C (P) (X) or Y. Specifically, when P is the Lindelof property and X is a locally-P metrizable space, we show that dim C (P)(X) =l(X)(N0) , where l (X) is the Lindelof number of X, and when P is countable compactness and X is a normal space, we show that Y = in iota beta X upsilon X where upsilon X is the Hewitt realcompactification of X.