Banach spaces and algebras of continuous functions

Abstract

this space becomes a real Banach space. Finally, provided with the usual multiplication of functions, this space becomes a real Banach algebra. The function which is identically one on X is the multiplicative identity, and we denote it by e. According to the Banach-Stone theorem [5],2 if X is compact, then the topology of X is reflected in the algebraic and metric structure of C(X) in the sense that if C(X) and C( Y) are equivalent as Banach spaces, then X is homeomorphic to Y. This general structural relation suggests particularization: what are the specific algebraic or metric properties of C(X) which correspond to specific topological features of X? Significant results have been obtained in this direction by Myers, Eilenberg, and others [1; 2; 3; 4]. This paper is drawn along these lines, and is concerned in particular with those spaces of continuous functions which are defined on the topological product of two compact Hausdorff spaces, or on compact Hausdorff spaces which contain such a product.3 The central problems are those of characterization; for the cases of products, and spaces which contain a product subspace, characterizations have been obtained. I have also obtained characterizations of continuous functions spaces over fiber spaces and fiber bundles, but the results are quite technical and not very revealing, and therefore are not of sufficient interest to include here.