ALMOST TRANSITIVITY IN $\mathcal{C}_0$ SPACES OF VECTOR-VALUED FUNCTIONS

Abstract

By means of M -structure and dimension theory, we generalize some known results and obtain some new ones on almost transitivity in C 0 (L, X). For instance, if X has the strong Banach-Stone property, then almost transitivity of C 0 (L, X) is divided into two weaker properties, one of them depending only on topological properties of L and the other being closely related to the covering dimensions of L and X. This leads to some non-trivial examples of almost transitive C 0 (L, X) spaces.

Convex-transitivity and function spaces

“…Our strategy for the case L ∞ s (X), which we mainly concentrate on, is to show that for a given x ∈ S X the constant function χ [0,1] x ∈ L ∞ s (X) is a big point and that…”

“…First we will show that for any x ∈ S X the function χ [0,1] x is a big point. By using Fact 2.2 it suffices to show that any simple function F ∈ S L ∞ s (X) (as in Fact 2.2) is contained in conv(G L ∞ s (X) (χ [0,1] x)).…”

“…By using Fact 2.2 it suffices to show that any simple function F ∈ S L ∞ s (X) (as in Fact 2.2) is contained in conv(G L ∞ s (X) (χ [0,1] x)). Let F = χ A 1 y 1 + · · · + χ A n y n , where {A k | 1 k n} is a measurable partition of [0, 1] and y 1 , .…”

“…Let us check next that for each F ∈ S L ∞ s (X) and x ∈ S X it holds that χ [0,1] x ∈ conv(G L ∞ s (X) (F )). To achieve this we employ a kind of sliding hump argument.…”

“…467-468]). The function χ [0,1] x can be approximated by convex combinations coming from conv(G L ∞ s (X) (F )) as follows:…”