Cone characterization of Grothendieck spaces and Banach spaces containing c 0

Abstract

In this article we study the embeddability of cones in a Banach space X . First we prove that c 0 is embeddable in X if and only if its positive cone c + 0 is embeddable in X and we study some properties of Banach spaces containing c 0 in the light of this result. So, unlike with the positive cone of 1 which is embeddable in any non-reflexive space, c + 0 has the same behavior as the whole space c 0 . In the second part of this article we give a characterization of Grothendieck spaces X according to the geometry of cones of X * . By these results we give a partial positive answer to a problem of J.H. Qiu concerning the geometry of cones.

A version of Kalton’s theorem for the space of regular operators

“…Suppose that T : c 0 → Y is a 0-cone isomorphism and A, B constants such that A||x|| ≤ ||T x|| ≤ B||x|| for each x ∈ c + 0 . Let y n = T (e n ) then we have that inf ||y n || > 0 and || n i=1 a i y i || ≤ B max{a i |i = 1, .., n} for each a i ≥ 0 and n ∈ N. This implies that || n i=1 a i y i || ≤ 2B max{|a i | | i = 1, .., n} for each a i ∈ R and n ∈ N(see [20], p.682). Since e n w − → 0 and T is bounded, we have that (y n ) is weakly null and by passing to a subsequence we can assume that (y n ) contains a basic sequence.…”

“…Note that in this case we have that ||T x|| ≤ B(||x + || + ||x − ||) ≤ 2B||x|| for each x ∈ X, hence T is automatically a bounded operator. In [20], the authors proved that if there exists a stronger type of "cone isomorphism" of c 0 into a Banach space Y , then c 0 is embeddable in Y . Below we show that the same holds if we assume the existence of a 0-cone isomorphism.…”

“…In this paper we use a different approach and we work directly on the space of regular operators. In particular by using the concept of cone isomorphism(see [20,22]) we are able to modify Kalton's main argument, in order to work for the regular operator norm. Moreover in our approach we use ideas and results from the work of Chen and Wickstead([10,11,12,23,24]) in the space of regular operators.…”

Grothendieck ordered Banach spaces with an interpolation property

“…Trivial examples of Grothendieck spaces are the reflexive spaces and of non-Grothendieck the non-reflexive, separable spaces. For some recent results on Grothendieck spaces, independent from this article, we refer to [10], [2] and [14]. In [14], Theorem 15, the following cone characterization is proved, which unfortunately cannot be applied, at least directly in this article.…”

“…For some recent results on Grothendieck spaces, independent from this article, we refer to [10], [2] and [14]. In [14], Theorem 15, the following cone characterization is proved, which unfortunately cannot be applied, at least directly in this article. A Banach space X is non-Grothendieck if and only if there exists a well-based cone P of X * such that int(P 0 ) = ∅ and the set of quasi-interior points of P 0 with respect to the…”