Spaces of Continuous Functions

Let X be a completely regular topological space, B(X) the Banach space of real-valued bounded continuous functions on X, with the usual norm ||&|| =supa?£x|&(#)| • A subset GCB(X) is called completely regular (c.r.) over X if given any closed subset KQ.X and point XoÇzX — K, there exists a ô £ G such that &(#o) = |NI a n ( i sup^^is: \b(x)\ <||&||. A topological space X is completely regular in the usual sense if and only if B(X) is c.r. over X. A Banach space B is said to act completely regularly (c.r.) on X if B is equivalent to a closed linear subspace of B(X) which is c.r. over X. I t is known [6 J that if X is compact, a closed linear subspace of B{X) c.r. over X determines the topology of X. By this is meant that if X\ and X2 are compact, and a Banach space B acts c.r. on both Xi and X2, then Xi is homeomorphic to X2. If B acts c.r. on X (compact or not), X is homeomorphically imbeddable in the surface of the unit sphere in 23*, the conjugate space to B under the weak-* topology, and for each 6£J3 and xÇzX we have the formula 6 ( x ) = i n f l G r [ | | 6 + / | | | H | ] , where r = {tGB\ t(x) = |MI}. If we weaken the definition of complete regularity so that G is c.r. over X means that for every closed set KC.X and point x0(E.X — K there is a 6 E G such that b(x0) = |HI> s u p ^ ^ b(x) <||&||, then a closed linear subspace of B(X) c.r. over X does not necessarily determine the topology of X. For example, if X consists of just two points, xi and X2, then the subspace G of B(X) consisting of all bÇ:B(X) such that b(xi) — —b(x2) is c.r. over X according to the weakened definition, yet it is equivalent to the space B(Y), where Y consists of a single point. Proper closed linear subspaces of B{X) which are c.r. over X exist in general for both compact and non-compact X, and may contain the constant functions. This is in contrast to the situation when B{X) is made into a normed ring (Banach algebra) R(X) or into a Banach lattice L(X); if X is compact, no proper closed subring of R(X) containing the constant functions can be c.r. over X [s]t and no proper closed sublattice of L(X) containing the constant functions can be c.r. over X [4]. Since topological properties of X must be reflected in metric and