On Certain Geometric Properties in Banach Spaces of Vector-Valued Functions

Abstract

We consider a certain type of geometric properties of Banach spaces, which includes, for instance, octahedrality, almost squareness, lushness and the Daugavet property. For this type of properties, we obtain a general reduction theorem, which, roughly speaking, states the following: if the property in question is stable under certain finite absolute sums (for example, finite p -sums), then it is also stable under the formation of corresponding Köthe-Bochner spaces (for example, L p -Bochner spaces). From this general theorem, we obtain as corollaries a number of new results as well as some alternative proofs of already known results concerning octahedral and almost square spaces and their relatives, diameter two properties, lush spaces and other classes.

A property in vector-valued function spaces

“…We will deal with the property ( * * ) for L ∞ (µ, X) and L 1 (µ, X). Very recently, it has been shown in [10,Theorem 4.8] that if X has the property ( * * ), then L 1 (µ, X) and L ∞ (µ, X) also have the property ( * * ). In fact, even more general reduction theorem is proved in [10] for a large class of spaces, such as octahedral and almost square spaces, lush spaces and so on.…”

“…Very recently, it has been shown in [10,Theorem 4.8] that if X has the property ( * * ), then L 1 (µ, X) and L ∞ (µ, X) also have the property ( * * ). In fact, even more general reduction theorem is proved in [10] for a large class of spaces, such as octahedral and almost square spaces, lush spaces and so on. However, we do not think the converse of the previous result, that is if L 1 (µ, X) or L ∞ (µ, X) has the property ( * * ), then so does X, can be deduced from this reduction theorem.…”

“…(3) Every space with the property ( * * ) has the MUP. Very recently, a stability results that X having the property ( * * ) implies that L 1 (µ, X) and L ∞ (µ, X) also have the the property ( * * ) with (Ω, Σ, µ) being a σfinite measure space has been proved in [10,Theorem 4.8] by a reduction theorem. In fact, this reduction theorem is shown in [10] for a large class of spaces that enjoy a certain type of geometric properties, such as octahedrality, almost squareness, lushness, the Daugavet property and so on.…”

“…Very recently, a stability results that X having the property ( * * ) implies that L 1 (µ, X) and L ∞ (µ, X) also have the the property ( * * ) with (Ω, Σ, µ) being a σfinite measure space has been proved in [10,Theorem 4.8] by a reduction theorem. In fact, this reduction theorem is shown in [10] for a large class of spaces that enjoy a certain type of geometric properties, such as octahedrality, almost squareness, lushness, the Daugavet property and so on. In the earlier time, stronger stability results for lushness have already been stated in recent monograph [13]: C(K, X) is lush if and only if X is, and the same results hold for L 1 (µ, X) and L ∞ (µ, X).…”