Abstract
Given a Banach space X, we define the number lambda(0)(X) = inf d(X-2,l(1)(2)), where the infimum is taken over all two-dimensional sub-spaces X2 of X. Here, d(M, N) means the Banach-Mazur distance between Banach spaces M,N defined by d(M, N) = inf {parallel to T parallel to parallel to T-1 parallel to : T : M -> N is an isomorphism}. We establish some facts about lambda(0) and then consider applications to Banach Stone type theorems for isomorphisms on continuous, vector -valued function spaces. If Q, K are locally compact Hausdorff spaces, and X,Y are Banach spaces for which both lambda(0)(X*) and lambda(0)(Y*) are greater than one, it has been shown that if T is an isomorphism from C-0(Q,E) onto C-0(K, Y) with parallel to T parallel to parallel to T-1 parallel to sufficiently small, then Q and K are homeomorphic, a generalization of the Banach Stone Theorem for isometrics. We examine such results for subspaces of these spaces. A closed subspace M of C-0(Q, X) is said to be a C-0(Q)-module if it is closed under multiplication by functions in C-0(Q). If M and N are C-0(Q), Co (K)-modules, respectively, then with assumptions similar to those mentioned above, we are able to obtain results in which the homeomorphism is between the strong boundaries of N and M. In this case, the strong boundaries are the subsets of K and Q, respectively, upon which the functions in N and M have nonzero values. We also obtain a new theorem concerning isometrics.