Isomorphisms of spaces of norm-continuous functions
If X is a compact Hausdorff space and E a Banach space, then C(X,E) denotes the space of continuous functions on X to E provided with the supremum norm. If E is the one-dimensional field of scalars then we write C(X) for C(X,E). The well known Banach-Stone theorem states that, for compact X and Y, if C(X) and C(Y) are isometric then X and Y are homeomorphic. Various authors, beginning with M. Jerison, have considered the problem of determining geometric properties of E which allow generalizations of this theorem to spaces of norm-continuous vector functions C(X,E). Another type of generalization of the theorem, while still dealing with scalar functions, replaces isometries by isomorphisms T with \(\| T\| \| T^{-1}\|\) small. In this article these two directions of generalization are combined. It is shown that if E is uniformly convex then the existence of an isomorpism T of C(X,E) onto C(Y,E), with \(\| T\| \| T^{- 1}\|\) less than a constant depending on the modulus of convexity associated with E, implies that X and Y are homeomorphic.
scientific article; zbMATH DE number 3909096