Disjointly homogeneous Banach lattices: Duality and complementation (original) (raw)
Disjointly homogeneous Banach lattices and applications
This is a survey on disjointly homogeneous Banach lattices and their applicactions. Several structural properties of this class are analyzed. In addition we show how these spaces provide a natural framework for studying the compactness of powers of operators allowing for a unified treatment of well-known results.
Disjointly homogeneous Banach lattices and compact products of operators
Journal of Mathematical Analysis and …, 2009
The notion of disjointly homogeneous Banach lattice is introduced. In these spaces every two disjoint sequences share equivalent subsequences. It is proved that on this class of Banach lattices the product of a regular AM-compact and a regular disjointly strictly singular operators is always a compact operator.
Disjointly non-singular operators on Banach lattices
Journal of Functional Analysis, 2021
An operator T from a Banach lattice E into a Banach space is disjointly non-singular (DN-S, for short) if no restriction of T to a subspace generated by a disjoint sequence is strictly singular. We obtain several results for DN-S operators, including a perturbative characterization. For E = L p (1 < p < ∞) we improve the results, and we show that the DN-S operators have a different behavior in the cases p = 2 and p = 2. As an application we prove that the strongly embedded subspaces of L p form an open subset in the set of all closed subspaces.
Revista Matemática Complutense, 2020
We show that c 0 is not a projective Banach lattice, answering a question of B. de Pagter and A. Wickstead. .On the other hand, we show that c 0 is complemented in the free Banach lattice generated by itself (seen as a Banach space). As a consequence, the free Banach lattice generated by c 0 is not projective.
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.
An Amir-Cambern theorem for subspaces of Banach lattice-valued continuous functions
arXiv (Cornell University), 2020
For i = 1, 2, let E i be a reflexive Banach lattice over R with a certain parameter λ + (E i) > 1, let K i be a locally compact (Hausdorff) topological space and let H i be a closed subspace of C 0 (K i , E i) such that each point of the Choquet boundary Ch H i K i of H i is a weak peak point. We show that if there exists an isomorphism T : H 1 → H 2 with T • T −1 < min{λ + (E 1), λ + (E 2)} such that T and T −1 preserve positivity, then Ch H 1 K 1 is homeomorphic to Ch H 2 K 2. 2010 Mathematics Subject Classification. 47B38; 46A55.
Journal of Soviet Mathematics, 1982
A survey is given of papers on the theory of normed and Banaeh lattices in the last 10 years; many directions of investigations on the theory of Banach spaces of measurable functions are illuminated.
Banach spaces with the 222-summing property
Transactions of the American Mathematical Society, 1995
A Banach space X has the 2-summing property if the norm of every linear operator from X to a Hilbert space is equal to the 2-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar field: the property is self-dual and any space with the property is a finite dimensional space of maximal distance to the Hilbert space of the same dimension. In the case of real scalars only the real line and real ℓ 2 ∞ have the 2-summing property. In the complex case there are more examples; e.g., all subspaces of complex ℓ 3 ∞ and their duals. 0. Introduction: Some important classical Banach spaces; in particular, C(K) spaces, L 1 spaces, the disk algebra; as well as some other spaces (such as quotients of L 1 spaces by reflexive subspaces [K], [Pi]), have the property that every (bounded, linear) operator from the space into a Hilbert space is 2-summing. (Later we review equivalent formulations of the definition of 2-summing operator. Here we mention only that an operator T : X → ℓ 2 is 2-summing provided that for all operators u : ℓ 2 → X the composition T u is a Hilbert-Schmidt operator; moreover, the 2-summing norm π 2 (T) of T is the supremum of the Hilbert-Schmidt norm of T u as u ranges over all norm one operators u : ℓ 2 → X.) In this paper we investigate the isometric version of this property: say that a Banach space X has the 2-summing property provided that π 2 (T) = T for all operators T : X → ℓ 2. While the 2-summing property is a purely Banach space concept and our investigation lies purely in the realm of Banach space theory, part of the motivation for studying the 2-summing property comes from operator spaces. In [Pa], Paulsen defines for a Banach space X the parameter α(X) to be the supremum of the completely bounded norm of T as T ranges over all norm one operators from X into the space B(ℓ 2) of all bounded linear operators on ℓ 2 and asks which spaces X have the property that α(X) = 1. Paulsen's problem and study of α(X) is motivated by old results of von Neumann, Sz.-Nagy, Arveson, and Parrott as well as more recent research of Misra and Sastry. The connection between Paulsen's problem and the present paper is Blecher's result [B] that α(X) = 1 implies that X has the 2-summing property. Another connection is through the property (P)
Geometry of unit balls of free Banach lattices, and its applications
arXiv (Cornell University), 2023
We begin by describing the unit ball of the free p-convex Banach lattice over a Banach space E (denoted by FBL (p) [E]) as a closed solid convex hull of an appropriate set. Based on it, we show that, if a Banach space E has the λ-Approximation Property, then FBL (p) [E] has the λ-Positive Approximation Property. Further, we show that operators u ∈ B(E, F) (where E and F are Banach spaces) which extend to lattice homomorphisms from FBL (q) [E] to FBL (p) [F ] are precisely those whose adjoints are (q, p)-mixing. Related results are also obtained for free lattices with an upper p-estimate. Contents 1. Introduction and preliminaries 2 2. Representing unit balls as solid convex hulls 4 2.1. Survey of solid convex hulls 4 2.2. The unit ball of FBL (p) [E] 4 2.3. The unit ball of FBL ↑p [E] 9 3. Positive Bounded Approximation Property 10 4. Duals of free Banach lattices 13 4.1. Sums of atoms in FBL ↑p [E] * 13 4.2. Sums of atoms in FBL (p) [E] * 14