Isomorphic vector-valued Banach-Stone theorems for subspaces
Tóm tắt
Given a Banach space X, we define the number λ0(X) = inf d(X2, ℓ1(2)), where the infimum is taken over all two-dimensional subspaces X2 of X. Here, d(M, N) means the Banach-Mazur distance between Banach spaces M, N defined by d(M, N) = inf{‖T‖ ‖T-1‖ : T: M → N is an isomorphism}. We establish some facts about λ0 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 λ0(X*) and λ0(Y*) are greater than one, it has been shown that if T is an isomorphism from C0(Q, E) onto C0(K, Y) with ‖T‖ ‖T-1 sufficiently small, then Q and K are homeomorphic, a generalization of the Banach-Stone Theorem for isometries. We examine such results for subspaces of these spaces. A closed subspace M of C0(Q, X) is said to be a C0(Q)-module if it is closed under multiplication by functions in C0(Q). If M and N are C0(Q), C0(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 isometries.
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