scholarly journals Geometric Characterization of Injective Banach Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 250
Author(s):  
Anatoly Kusraev ◽  
Semën Kutateladze
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Set Theory ◽  
Banach Lattice ◽  
Dual Space ◽  
Banach Lattices ◽  
Transfer Principle ◽  
Ordered Banach Spaces

This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach lattice. The study bases on the Boolean valued transfer principle for injective Banach lattices. The latter states that each such lattice serves as an interpretation of an AL-space in an appropriate Boolean valued model of set theory. External identification of the internal Boolean valued properties of the corresponding AL-spaces yields a characterization of injective Banach lattices among Banach spaces and ordered Banach spaces. We also describe the structure of the dual space and present some dual characterization of injective Banach lattices.

scholarly journals Banach spaces with property (w)

1993 ◽  
Vol 35 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Weakly Compact ◽  
Type Spaces

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

scholarly journals Monotonic norms in ordered Banach spaces

1988 ◽  
Vol 45 (2) ◽  
pp. 217-219 ◽  
Author(s):  
K. F. Ng ◽  
C. K. Law
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Alternative Proof ◽  
Ordered Banach Space ◽  
Dual Result ◽  
Ordered Banach Spaces

AbstractLet B be an ordered Banach space with ordered Banach dual space. Let N denote the canonical half-norm. We give an alternative proof of the following theorem of Robinson and Yamamuro: the norm on B is α-monotone (α ≥ 1) if and only if for each f in B* there exists g ∈ B* with g ≥ 0, f and ∥g∥ ≤ α N(f). We also establish a dual result characterizing α-monotonicity of B*.

scholarly journals Strongly exposed points and a characterization of l1 (Γ) by the Schur property

1987 ◽  
Vol 30 (3) ◽  
pp. 397-400 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Spaces ◽  
Banach Lattice ◽  
Positive Cone ◽  
Schur Property ◽  
Positive Elements ◽  
Strongly Exposed Points ◽  
Ordered Banach Spaces ◽  
Convex Subsets ◽  
Quasi Interior

In this paper we study the existence of strongly exposed points in unbounded closed and convex subsets of the positive cone of ordered Banach spaces and we prove the following characterization for the space l1(Γ): A Banach lattice X is order-isomorphic to l1(Γ) iff X has the Schur property and X* has quasi-interior positive elements.

scholarly journals Extension of Riesz homomorphisms. III

1987 ◽  
Vol 43 (1) ◽  
pp. 35-46
Author(s):  
Gerard Buskes
Keyword(s):  
Banach Spaces ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Topological Spaces ◽  
Simultaneous Extension ◽  
Continuous Mappings

AbstractIn this paper we prove an analogue of the separable version of Nachbin's characterization of injective Banach spaces in the setting of Banach lattices. The mappings involved are continuous Riesz homomorphisms defined on ideals of separable Banach lattices which can be extended to Riesz homomorphisms on the whole Banach lattice. We discuss applications to simultaneous extension operators and to extension of continuous mappings between certain topological spaces.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions

1977 ◽  
Vol 29 (5) ◽  
pp. 963-970 ◽  
Author(s):  
Mark A. Smith
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Nonzero Element ◽  
Bounded Subset ◽  
Compact Sets ◽  
Weakly Compact ◽  
Uniform Rotundity ◽  
Minimum Depth ◽  
Weakly Compact Sets

In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

scholarly journals Semi-embeddings of Banach spaces which are hereditarily c0

1983 ◽  
Vol 26 (2) ◽  
pp. 163-167 ◽  
Author(s):  
L. Drewnowski
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Closed Unit Ball ◽  
Continuous Linear ◽  
Scattered Space ◽  
Complete Set

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).

scholarly journals Complex extreme measurable selections

1995 ◽  
Vol 58 (2) ◽  
pp. 222-231
Author(s):  
Douglas Mupasiri
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Extreme Point ◽  
Measure Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Closed Unit Ball ◽  
Measurable Selections ◽  
Necessary And Sufficient

AbstractWe give a characterization of complex extreme measurable selections for a suitable set-valued map. We use this result to obtain necessary and sufficient conditions for a function to be a complex extreme point of the closed unit ball of Lp (ω, Σ, ν X), where (ω, σ, ν) is any positive, complete measure space, X is a separable complex Banach space, and 0 < p < ∞.

scholarly journals On the canonical projection of the third dual of a Banach space onto the first dual

1976 ◽  
Vol 15 (3) ◽  
pp. 351-354 ◽  
Author(s):  
A.L. Brown
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Canonical Projection ◽  
The Third ◽  
Current Belief ◽  
A Current

For a Banach space B let P denote the canonical projection of the third dual space of B onto the embedding of the first dual into the third. It is shown that if B = l1 then ‖I-P‖ = 2.This fact shows to be mistaken a current belief in a statement which is equivalent to the statement that for all Banach spaces B the operator I – P is of norm one.

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