Convex-transitivity and function spaces

Abstract

It is shown that if X is a convex-transitive Banach space and 1 p < ∞, then L p ([0, 1], X) and L ∞ s ([0, 1], X) are convextransitive. Here L ∞ s ([0, 1], X) is the closed linear span of the simple functions in the Bochner space L ∞ ([0, 1], X). If H is an infinite-dimensional Hilbert space and C 0 (L) is convex-transitive, then C 0 (L, H) is convex-transitive. Some new fairly concrete examples of convex-transitive spaces are provided.

Convex-Transitivity of Banach Algebras via Ideals

“…Indeed, X is isometric to the 1-codimensional subspace of L 1 of all the functions with average 0. It was pointed out in [22] that this subspace is in turn almost transitive, even though it is neither isometric to L 1 , nor contractively complemented in it.…”

“…We note that the technical homogeneity property on I is motivated by considerations in [19]. For example, the condition holds if I is the ideal of sets of cardinality < κ, or, even for n = 1, if I is the dual ideal of an ultrafilter, which does not contain any sets of cardinality < κ (see the proof of Lemma 3.2 in [22]). If κ = ω, then the above result contains the result appearing in [19], namely that ℓ ∞ (X)/c 0 (X) is convex-transitive for uniformly convex-transitive X.…”

Convex-transitive Douglas algebras

“…Here we provide examples of concrete ω * -convex-transitive complex Banach algebras modeled on the unit disk. Specimens of convex-transitive spaces can be found for example in [7], [8], [24], [34], [35], [38], and [39].…”

Uniformly Convex-Transitive Function Spaces

“…Greim, Jamison and Kaminska [11] proved that if X is almost transitive and 1 ≤ p < ∞, then the Lebesgue-Bochner space L p (X) is also almost transitive. Recently, an analogous study of the spaces C 0 (L, X) was done by Aizpuru and Rambla [1], and some related spaces were studied by Talponen [19]. For some other relevant contemporary results, see [5], [13] and [16].…”