Isomorphic Banach-Stone theorems and isomorphisms which are close to isometries

Ehrhard Behrends

Vol. 133 (1988), No. 2, 229–250

DOI: 10.2140/pjm.1988.133.229

Abstract

A Banach space X is said to have the isomorphic Banach-Stone property if for locally compact Hausdorff spaces K and L one always can conclude that K and L are homeomorphic provided that the Banach spaces C₀(K,X) and C₀(L,X) (=the continuous X-valued functions on K resp. L which vanish at infinity) are isomorphic with sufficiently small Banach-Mazur distance.

Our main results are a characterization of the finite-dimensional spaces with this property, and we also get an abundance of new finite- and infinite-dimensional examples.

These results appear as corollaries to general theorems about isomorphisms between certain spaces of continuous vector-valued functions. They enable us also to conclude that, for certain spaces X, and all compact K all isomorphisms T on C₀(K,X) with (1∕(1 + τ))∥f∥≤∥Tf∥≤ (1 + τ)∥f∥ for small τ can be approximated by isometries.

Mathematical Subject Classification 2000

Primary: 46B20

Milestones

Received: 29 April 1987

Revised: 10 August 1987

Published: 1 June 1988