A Banach space not containing l1l_{1}l1 whose dual ball is not weak${}^ ast$ sequentially compact (original) (raw)
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Reflexivity and the Separable Quotient Problem for a Class of Banach Spaces
Bulletin of the Polish Academy of Sciences Mathematics
Let E be a Banach lattice and let X be its closed subspace such that: X is complemented in E, or the norm of E is order continuous. Then X is reflexive iff X* contains no isomorphic copy of \ell_1 iff for every n ≥ 1, the nth dual X^(n) of X contains no isomorphic copy of \ell_1 iff X has no quotient isomorphic to c_0 and X does not have a subspace isomorphic to \ell_1 (Theorem 2). This is an extension of the results obtained earlier by Lozanovski˘ i, Tzafriri, Meyer-Nieberg, and Diaz and Fern´andez. The theorem is applied to show that many Banach spaces possess separable quotients isomorphic to one of the following spaces: c_0, \ell_1, or a reflexive space with a Schauder basis.
On Banach spaces whose dual balls are not weak∗ sequentially compact
Israel Journal of Mathematics, 1977
THEOREM 1. Let X be a Banach space. (a) If X* has a closed subspace in which no normalized sequence converges weak* to zero, then I, is isomorphic to a subspace of X. (b) If X* contains a bounded sequence which has no weak* convergent subsequence, then X contains a separable subspace whose dual is not separable.
Observations on the Separable Quotient Problem for Banach Spaces
Axioms
The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space E * onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in E * . It is shown that every dual-like Banach space has an infinite-dimensional separable quotient.
On dual of Banach sequence spaces
J. Hagler and P. Azimi have introduced a class of Banach sequence spaces, the X α,1 spaces as a class of hereditarily 1 Banach spaces. In this paper, we show that (i) X * α,1 , the dual of Banach space X α,1 contains asymptotically isometric copies of ∞, (ii) X * α,1 is nonseparable although X α,1 is a separable Banach space. Also, we show X α,1 is not hereditarily indecomposable. p. Here, using two methods we show that the Banach spaces X * α,1 , the dual of Banach spaces X α,1 , are nonseparable. By the first method, we show X * α,1 contain asymptotically isometric copy of ∞. A result of [6] shows that X * α,1 contain isometric copy of ∞ , and then they are nonseparable. By the second method,
The structure of nonseparable Banach spaces with uncountable unconditional bases
Let X be a Banach space with an uncountable unconditional Schauder basis, and let Y be an arbitrary nonseparable subspace of X. If X contains no isomorphic copy of 1(J ) with J uncountable then (1) the density of Y and the weak*-density of Y * are equal, and (2) the unit ball of X * is weak* sequentially compact. Moreover, (1) implies that Y contains large subsets consisting of pairwise disjoint elements, and a similar property holds for uncountable unconditional basic sets in X.
On separably injective Banach spaces
Advances in Mathematics, 2013
In this paper we deal with two weaker forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. We show several structural and stability properties of these classes of Banach spaces. We provide natural examples of (universally) separably injective spaces, including L∞ ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We obtain two fundamental characterizations of universally separably injective spaces: a) A Banach space E is universally separably injective if and only if every separable subspace is contained in a copy of ℓ∞ inside E. b) A Banach space E is universally separably injective if and only if for every separable space S one has Ext(ℓ∞/S, E) = 0. The final Section of the paper focuses on special properties of 1-separably injective spaces. Lindenstrauss obtained in the middle sixties a result that can be understood as a proof that, under the continuum hypothesis, 1-separably injective spaces are 1-universally separably injective; he left open the question in ZFC. We construct a consistent example of a Banach space of type C(K) which is 1-separably injective but not 1-universally separably injective.
Israel Journal of Mathematics, 2009
A Banach space X will be called extensible if every operator E → X from a subspace E ⊂ X can be extended to an operator X → X. Denote by dens X. The smallest cardinal of a subset of X whose linear span is dense in X, the space X will be called automorphic when for every subspace E ⊂ X every into isomorphism T : E → X for which dens X/E = dens X/T E can be extended to an automorphism X → X. Lindenstrauss and Rosenthal proved that c 0 is automorphic and conjectured that c 0 and 2 are the only separable automorphic spaces. Moreover, they ask about the extensible or automorphic character of c 0 (Γ), for Γ uncountable. That c 0 (Γ) is extensible was proved by Johnson and Zippin, and we prove here that it is automorphic and that, moreover, every automorphic space is extensible while the converse fails. We then study the local structure of extensible spaces, showing in particular that an infinite dimensional extensible space cannot contain uniformly complemented copies of n p , 1 ≤ p < ∞, p = 2. We derive that infinite dimensional spaces such as Lp(µ), p = 2, C(K) spaces not isomorphic to c 0 for K metric compact, subspaces of c 0 which are not isomorphic to c 0 , the Gurarij space, Tsirelson spaces or the Argyros-Deliyanni HI space cannot be automorphic.
Separable quotients of Banach spaces
1997
In this survey we shaw that the separable quotient problem for Banach spac~in equivalent to several other problema from Banach space theory. Wc give afro several partial solutiona to the prob1cm.