Uniquely generated Grothendieck space ideals

1985 ◽  
Vol 99 (3) ◽  
pp. 235-244 ◽  
Author(s):  
Marilda A. Sim�es
Keyword(s):  
Axioms ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 34 ◽  
Author(s):  
Juan Carlos Ferrando ◽  
Salvador López-Alfonso ◽  
Manuel López-Pellicer

We call a subset M of an algebra of sets A a Grothendieck set for the Banach space b a ( A ) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence μ n n = 1 ∞ in b a ( A ) which is pointwise convergent on M is weakly convergent in b a ( A ) , i.e., if there is μ ∈ b a A such that μ n A → μ A for every A ∈ M then μ n → μ weakly in b a ( A ) . A subset M of an algebra of sets A is called a Nikodým set for b a ( A ) if each sequence μ n n = 1 ∞ in b a ( A ) which is pointwise bounded on M is bounded in b a ( A ) . We prove that if Σ is a σ -algebra of subsets of a set Ω which is covered by an increasing sequence Σ n : n ∈ N of subsets of Σ there exists p ∈ N such that Σ p is a Grothendieck set for b a ( A ) . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a σ -algebra Σ is covered by an increasing sequence Σ n : n ∈ N of subsets, there is p ∈ N such that Σ p is a Nikodým set for b a Σ . This also refines the Grothendieck result stating that for each σ -algebra Σ the Banach space ℓ ∞ Σ is a Grothendieck space. Some applications to classic Banach space theory are given.


Author(s):  
Aníbal Moltó

SynopsisIn this paper, a class of Boolean rings containing the class discussed in papers by Seever (1968) and Faires (1976), is defined in such a way that an extension of the classical Vitali–Hahn–Saks theorem holds for exhausting additive set functions. Some new compact topological spaces K for which C(K) is a Grothendieck space are constructed and a Nikodym type theorem is deduced from it. The Boolean algebras of Seever and Faires and those we study here are defined by ‘interpolation properties’ between disjoint sequences in the algebra. We give an example at the end of the paper that illustrates the difficulties arising when we try to find a larger class of Boolean algebras, defined in terms of such properties, for which the Vitali–Hanh–Saks theorem holds.


Author(s):  
Werner Ricker

AbstractIt is shown that in a Grothendieck space with the Dunford-Pettis property, the class of well-bounded operators of type (B) coincides with the class of scalar-type spectral operators with real spectrum. It turns out that in such Banach spaces, analogues of the classical theorems of Hille-Sz. Nagy and Stone concerned with the integral representation of C0-semigroups of normal operators and strongly continuous unitary groups in Hilbert spaces, respectively, are of a very special nature.


1989 ◽  
Vol 39 (3) ◽  
pp. 321-328 ◽  
Author(s):  
Richard Becker

Let B be a Banach space. consider the convex proper weakly complete cones X contained in B′ with σ(B′, B) such that X ∩ B′, is conic in the sense of Asimow: that is, there exists α ≥ 0 and f ∈ B″ such that ‖ ‖B ≤ f ≤ α·‖ ‖B on X. This class arises in the theory of integral representations.If B is reflexive, such a cone has a weakly-compact basis. This paper considers the converse problem:- if one requires that X ∩ B′1 be σ(B′, B) metrisable, the existence of X (without a compact σ(B′, B) basis) is equivalent to the statement that B is not a Grothendieck space.However, in every space C(K) with infinitely compact K, one can find such a cone X. If two such cones in B′ are not too far apart, their sum belongs to this class.


2012 ◽  
Vol 55 (3) ◽  
pp. 449-461 ◽  
Author(s):  
Manijeh Bahreini ◽  
Elizabeth Bator ◽  
Ioana Ghenciu

AbstractWe study the complementation of the space W(X, Y) of weakly compact operators, the space K(X, Y) of compact operators, the space U(X, Y) of unconditionally converging operators, and the space CC(X, Y) of completely continuous operators in the space L(X, Y) of bounded linear operators from X to Y. Feder proved that if X is infinite-dimensional and c0 ↪ Y, then K(X, Y) is uncomplemented in L(X, Y). Emmanuele and John showed that if c0 ↪ K(X, Y), then K(X, Y) is uncomplemented in L(X, Y). Bator and Lewis showed that if X is not a Grothendieck space and c0 ↪ Y, then W(X, Y) is uncomplemented in L(X, Y). In this paper, classical results of Kalton and separably determined operator ideals with property (∗) are used to obtain complementation results that yield these theorems as corollaries.


1983 ◽  
Vol 75 (2) ◽  
pp. 193-216 ◽  
Author(s):  
J. Bourgain
Keyword(s):  

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