Extreme Point Methods and Banach-Stone Theorems

Abstract

An operator is said to be nice if its conjugate maps extreme points of the dual unit ball to extreme points. The classical Banach-Stone Theorem says that an isometry from a space of continuous functions on a compact Hausdorff space onto another such space is a weighted composition operator. One common proof of this result uses the fact that an isometry is a nice operator. We use extreme point methods and the notion of centralizer to characterize nice operators as operator weighted compositions on subspaces of spaces of continuous functions with values in a Banach space. Previous characterizations of isometries from a subspace M of Q,( Q, X) into Q>(K, Y) require Y to be strictly convex, but we are able to obtain some results without that assumption. Important use is made of a vector-valued version of the Choquet Boundary. We also characterize nice operators from one function module to another.2000 Mathematics subject classification: primary 46B04, 46E40.

Linear Surjective Isometries Between Vector-Valued Function Spaces

“…The result (1) follows the same line of reasoning as previous results [1,6,18,25,28]. On the other hand, (2) and (3) seem to exhibit a relatively new variant in that their assumptions are involved in the topology of underlying spaces X and Y (see [33] and a paper by K. Kawamura and Miura, 'Real-linear surjective isometries between function spaces,' which has been submitted for publication).…”

Nice Operators into G-Spaces

“…Such operators have been studied in the literature. See [1,2,4,7,10,14,15,21,22,24,27,29,30] and [25] for more recent results.…”

Extreme contractions on continuous vector-valued function spaces

“…The general question to be asked, then, is the converse: if T ∈ L(X, Y ) is extreme, must it be nice? The answer, in general, is no, since it has been shown that there exists a four-dimensional Banach space X and an extreme contraction T from X to C [0,1] that is not nice [2]. However, there is a positive answer for certain C(K) spaces, as we have already mentioned above, and other versions of the BLP result have been given by several authors.…”

“…This form is typical of nice operators on continuous function spaces, even in the vector-valued case [1].…”