Normality of spaces of operators and quasi-lattices
TL;DR: In this paper, the authors give an overview of normality and conormality properties of pre-ordered Banach spaces and define a class of ordered spaces called quasi-lattices which strictly contain the Banach lattices, and prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi lattice.
Abstract: We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces \(X\) and \(Y\) with closed cones we investigate normality of \(B(X,Y)\) in terms of normality and conormality of the underlying spaces \(X\) and \(Y\). Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples \(X\) and \(Y\) that are not Banach lattices, but for which \(B(X,Y)\) is normal. In particular, we show that a Hilbert space \(\mathcal {H}\) endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if \(\dim \mathcal {H}\ge 3\)), and satisfies an identity analogous to the elementary Banach lattice identity \(\Vert |x|\Vert =\Vert x\Vert \) which holds for all elements \(x\) of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.
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TL;DR: Blecher and Read as mentioned in this paper introduced a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algesbras.
Abstract: Blecher and Read have recently introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. In the present paper we generalize some part of this, and some other facts, to larger classes of Banach algebras.
22 citations
Cites background from "Normality of spaces of operators an..."
...In the language of [40], the last result implies that the associated preorder on A there is 1-absolutely conormal....
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TL;DR: Solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping.
Abstract: In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others.
14 citations
07 Sep 2016
TL;DR: In this paper, the vertical and lateral extensions of the Bochner integral have been studied for functions with values in a partially ordered vector space, and the vertical extension has been compared with other integrals.
Abstract: This thesis consists of two distinct topics. The first part of the thesis con- siders Gibbs-non-Gibbs transitions. Gibbs measures describe the macro- scopic state of a system of a large number of components that is in equilib- rium. It may happen that when the system is transformed, for example, by a stochastic dynamics that runs over a certain time interval, the evolved state is no longer a Gibbs measure. We study transitions from Gibbs tot non-Gibbs for mean-field systems and their relation to the large deviation rate function that is related to those systems. In the second part of the thesis we describe different notions of integrals for functions with values in a partially ordered vector space. We describe two extensions for integrals, called the vertical and the lateral extension. We compare combinations of them and compare them to other integrals. Another integral can be obtained for Archimedean directed ordered vector spaces, as they can be covered by Banach spaces in a natural way. This allows us to generalise the Bochner integral to function with values in such space.
7 citations
Cites background from "Normality of spaces of operators an..."
...3Batty and Robinson [4] call the cone D+ approximately C-absolutely dominating, and, Messerschmidt [68] calls D approximately C-absolutely conormal , if the norm on D is C-absolutely dominating....
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TL;DR: In this paper, the geometric duality theory of real pre-ordered Banach spaces is extended to real Banach space endowed with arbitrary collections of closed cones, and geometric notions of normality, conormality, additivity and coadditivity for members of dual pairs of real vector spaces are defined as certain possible interactions between two cones and two convex sets containing zero.
6 citations
Posted Content•
TL;DR: In this paper, the authors generalize the work of Werner and others to develop two abstract characterizations for self-adjoint operator spaces, and demonstrate a generalization of the Arveson Extension Theorem in this context.
Abstract: In this paper, we generalize the work of Werner and others to develop two abstract characterizations for self-adjoint operator spaces. The corresponding abstract objects can be represented as self-adjoint subspaces of $B(H)$ in such a way that both a metric structure and an order structure are preserved at each matrix level. We demonstrate a generalization of the Arveson Extension Theorem in this context. We also show that quotients of self-adjoint operator spaces can be endowed with a compatible operator space structure and characterize the kernels of completely positive completely bounded maps on self-adjoint operator spaces.
6 citations
Cites background from "Normality of spaces of operators an..."
...zk}. For brevity, we call a normed ordered vector space which is normal a normal orderedvector space. Normal ordered vector spaces are sometimes called 1-normal or 1-max-normal in the literature. See [12] for an overview of this terminology. We will not be considering other notions of normality here, so we have decided to simplify the terminology. The following Proposition shows that ordered vector sp...
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References
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Book•
09 Oct 1998
TL;DR: In this article, the Hahn-Banach Extension Theorem (HBMT) is used to describe the properties of normed spaces and linear operators between normed space.
Abstract: 1 Basic Concepts.- 1.1 Preliminaries.- 1.2 Norms.- 1.3 First Properties of Normed Spaces.- 1.4 Linear Operators Between Normed Spaces.- 1.5 Baire Category.- 1.6 Three Fundamental Theorems.- 1.7 Quotient Spaces.- 1.8 Direct Sums.- 1.9 The Hahn-Banach Extension Theorems.- 1.10 Dual Spaces.- 1.11 The Second Dual and Reflexivity.- 1.12 Separability.- 1.13 Characterizations of Reflexivity.- 2 The Weak and Weak Topologies.- 2.1 Topology and Nets.- 2.2 Vector Topologies.- 2.3 Metrizable Vector Topologies.- 2.4 Topologies Induced by Families of Functions.- 2.5 The Weak Topology.- 2.6 The Weak Topology.- 2.7 The Bounded Weak Topology.- 2.8 Weak Compactness.- 2.9 James's Weak Compactness Theorem.- 2.10 Extreme Points.- 2.11 Support Points and Subreflexivity.- 3 Linear Operators.- 3.1 Adjoint Operators.- 3.2 Projections and Complemented Subspaces.- 3.3 Banach Algebras and Spectra.- 3.4 Compact Operators.- 3.5 Weakly Compact Operators.- 4 Schauder Bases.- 4.1 First Properties of Schauder Bases.- 4.2 Unconditional Bases.- 4.3 Equivalent Bases.- 4.4 Bases and Duality.- 4.5 James's Space J.- 5 Rotundity and Smoothness.- 5.1 Rotundity.- 5.2 Uniform Rotundity.- 5.3 Generalizations of Uniform Rotundity.- 5.4 Smoothness.- 5.5 Uniform Smoothness.- 5.6 Generalizations of Uniform Smoothness.- A Prerequisites.- B Metric Spaces.- D Ultranets.- References.- List of Symbols.
1,099 citations
Book•
01 Dec 1972
TL;DR: In this article, the authors propose a method to solve the problem of convex control problems in Banach spaces. But this method is not suitable for functional analysis.Convex Functions and Convex Programming
Abstract: Fundamentals of Functional Analysis.- Convex Functions.- Convex Programming.- Convex Control Problems in Banach Spaces.
853 citations