A Characterization of C-Spaces

Abstract

If H is any compact Hausdorff space-that is, a Hausdorff space in which any arbitrary covering of the space by open sets can be reduced to a covering by a finite number of them-the aggregate C(H) of all real-valued continuous functions x(r), where r runs over II, is of interest from any one of several viewpoints. C(H) is a real Banach space with the natural notions of addition of two of its members, and multiplication by a real scalar; the norm of any element X(T) is defined by the maximum of I X(7) I for r E H. It may also be considered as a commutative normed ring with scalars, the operation of multiplication being again the usual one of multiplication of two functions. And from a third point of view it is a linear lattice, the partial ordering in this case being the one which again naturally suggests itself: x(r) < y(r) if this relationship is true in its elementary sense for every r e H. From each of these three aspects the systems C(H) are quite special. Not every Banach space is equivalent, in the sense in which that term is naturally defined for such spaces, to such a system 0(H), and the same is true of rings and lattices. The question arises, then, to determine, in each of these three classes, which ones of their members are so representable.' In the case of rings and lattices the answer has been given: necessary and sufficient conditions are known in order that a system of either of these two kinds shall be equivalent, in a natural and specified sense, to such a system C(H). For the class of Banach spaces the question has received some attention2 but it is believed that no answer has as yet appeared.3 Our object in the present paper is to furnish an answer to this question: we determine necessary and sufficient conditions on a real Banach space B in order that it shall be a C-space-in other words, that there shall exist a compact Hausdorff space H such that B is equivalent to C(H) in the sense that it is mappable onto 0(H) by a linear isomorphism which preserves the norm. It should be noticed that the apparent gain in generality obtained by considering more general spaces is not a real one. We might consider instead of a compact H, for example, a more general topological space T-say a T, space;