Hahn-Banach Type Research Articles

Hilbert C∗$C^*$‐module independence

AbstractWe introduce the notion of Hilbert ‐module independence: Let be a unital ‐algebra and let , be ternary subspaces of a Hilbert ‐module . Then, and are said to be Hilbert ‐module independent if there are positive constants m and M such that for every state on , there exists a state φ on such that We show that it is a natural generalization of the notion of ‐independence of ‐algebras. Moreover, we demonstrate that even in the case of ‐algebras, this concept of independence is new and has a nice characterization in terms of Hahn–Banach–type extensions. We show that if has the quasi extension property and with , then . Several characterizations of Hilbert ‐module independence and a new characterization of ‐independence are given. One of characterizations states that if is such that , then and are Hilbert ‐module independent if and only if for all and . We also provide some technical examples and counterexamples to illustrate our results.

On Hahn-Banach theorem and some of its applications

Abstract First, this work provides an overview of some of the Hahn-Banach type theorems. Of note, some of these extension results for linear operators found recent applications to isotonicity of convex operators on a convex cone. Next, the work investigates applications of the Krein-Milman theorem to representation theory and elements of Choquet theory. A sandwich theorem of intercalating an affine function h h between f f and g , g, where f f\hspace{.25em} and – g \mbox{--}g are convex, f ≤ g f\le g on a finite-simplicial set, is recalled. Its recent topological version is also noted. Here, the novelty is that a finite-simplicial set may be unbounded in any locally convex topology on the domain space. Third, the paper summarizes and comments on recently published applications of a Hahn-Banach extension result for positive linear operators, combined with polynomial approximation on unbounded subsets, to the Markov moment problem. Some applications to concrete spaces are detailed as well. Finally, this work provides a characterization of a finite-dimensional convex bounded subset in terms of the property that any convex function defined on that subset is bounded below. This last property remains valid for a large class of convex operators.

Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces

<abstract> <p>In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasinormed spaces.</p> </abstract>

On Markov Moment Problem, Polynomial Approximation on Unbounded Subsets, and Mazur–Orlicz Theorem

We review earlier and recent results on the Markov moment problem and related polynomial approximation on unbounded subsets. Such results allow proving the existence and uniqueness of the solutions for some Markov moment problems. This is the first aim of the paper. Our solutions have a codomain space a commutative algebra of (linear) symmetric operators acting from the entire real or complex Hilbert space H to H; this algebra of operators is also an order complete Banach lattice. In particular, Hahn–Banach type theorems for the extension of linear operators having a codomain such a space can be applied. The truncated moment problem is briefly discussed by means of reference citations. This is the second purpose of the paper. In the end, a general extension theorem for linear operators with two constraints is recalled and applied to concrete spaces. Here polynomial approximation plays no role. This is the third aim of this work.

From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications

The aim of this review paper is to recall known solutions for two Markov moment problems, which can be formulated as Hahn–Banach extension theorems, in order to emphasize their relationship with the following problems: (1) pointing out a previously published sandwich theorem of the type f ≤ h ≤ g, where f, −g are convex functionals and h is an affine functional, over a finite-simplicial set X, and proving a topological version for this result; (2) characterizing isotonicity of convex operators over arbitrary convex cones; giving a sharp direct proof for one of the generalizations of Hahn–Banach theorem applied to the isotonicity; (3) extending inequalities assumed to be valid on a small subset, to the entire positive cone of the domain space, via Krein–Milman or Carathéodory’s theorem. Thus, we point out some earlier, as well as new applications of the Hahn–Banach type theorems, emphasizing the topological versions of these applications.

Linear Hahn-Banach type extension operators in Banach algebras of operators

Linear Hahn-Banach type extension operators in Banach algebras of operators

Linear Hahn-Banach type extension operators in Banach algebras of operators

Linear Hahn-Banach type extension operators in Banach algebras of operators

Weak topologies for modules over rings of bounded random variables

In order to establish a functional analytic basis for representation theorems for conditional and multi-period risk measures, we study locally convex modules over the ring λ=L∞(G). Their topology is determined by λ-seminorms. As expected, central mathematical tools of the analysis are Hahn–Banach type and separation theorems which however have to be treated more carefully in the module case. Once a dual λ-module is introduced, one can establish a module version of the Bipolar theorem. We also prove the Krein–Šmulian as well as the Alaoglu–Bourbaki theorem for λ-modules. For Banach λ-modules their reflexivity is characterized by a compactness criterium in a (very) weak-* topology.

An interpolation problem for completely positive maps on matrix algebras: solvability and parametrization

We present certain existence criteria and parameterizations for an interpolation problem for completely positive maps that take given matrices from a finite set into prescribed matrices. Our approach uses density matrices associated to linear functionals on -subspaces of matrices, inspired by the Smith-Ward linear functional and Arveson’s Hahn-Banach Type Theorem. A necessary and sufficient condition for the existence of solutions and a parametrization of the set of all solutions of the interpolation problem in terms of a closed and convex set of an affine space are obtained. Other linear affine restrictions, like trace preserving, can be included as well, hence covering applications to quantum channels that yield certain quantum states at prescribed quantum states. We also perform a careful investigation on the intricate relation between the positivity of the density matrix and the positivity of the corresponding linear functional.

Some problems in functional analysis inspired by Hahn-Banach type theorems

As a cornerstone of functional analysis, Hahn-Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications of the Hahn-Banach theorem which are less familiar to the mathematical community, apart from highlighting certain aspects of the Hahn-Banach phenomena which have spurred intense research activity over the past few years, especially involving operator analogues and nonlinear variants of this theorem. For a discussion of a whole lot of issues related to the Hahn-Banach theorem not treated in this paper, the best source is a famous survey paper by Narici and Beckenstein [31] which deals, among other things, with the different settings witnessing the validity of the Hahn-Banach theorem.

Some new versions of Hahn-Banach theorem

In 1997, L.-G. Huang and X. Zhang introduced the cone metric space, which is the generalization of the metric space. Further, some authors introduced the cone normed space. In this paper, we introduce the conceptions of bounded linear operator from a normed space to a cone normed space and bounded linear operator between two cone normed spaces. A general Hahn-Banach type linear operator extension theorem from a linear space to a partial ordering linear space, with the order is induced by a minimal cone, is proven. Two preserved-norm bounded linear operator extension theorems, one is from a normed space to a cone normed space and another is between two cone normed spaces, and some corollaries are given.

Metric version of flatness and Hahn–Banach type theorems for normed modules over sequence algebras

We introduce and study the metric or extreme versions of the notions of a flat and an injective normed module. The relevant definitions, in contrast with the standard known ones, take into account the exact value of the norm of the module. The main result

Polynomials on Riesz spaces

In this paper we investigate polynomial mappings on Riesz spaces. We give a characterization of positivity of homogeneous polynomials in terms of forward differences. Finally we prove Hahn–Banach type extension theorems for positive and regular polynomial mappings.

Semantic Domains for Combining Probability and Non-Determinism

We present domain-theoretic models that support both probabilistic and nondeterministic choice. In [A. McIver and C. Morgan. Partial correctness for probablistic demonic programs. Theoretical Computer Science , 266:513–541, 2001], Morgan and McIver developed an ad hoc semantics for a simple imperative language with both probabilistic and nondeterministic choice operators over a discrete state space, using domain-theoretic tools. We present a model also using domain theory in the sense of D.S. Scott (see e.g. [G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott. Continuous Lattices and Domains , volume 93 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 2003]), but built over quite general continuous domains instead of discrete state spaces. Our construction combines the well-known domains modelling nondeterminism – the lower, upper and convex powerdomains, with the probabilistic powerdomain of Jones and Plotkin [C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the Fourth Annual Symposium on Logic in Computer Science , pages 186–195. IEEE Computer Society Press, 1989] modelling probabilistic choice. The results are variants of the upper, lower and convex powerdomains over the probabilistic powerdomain (see Chapter 4). In order to prove the desired universal equational properties of these combined powerdomains, we develop sandwich and separation theorems of Hahn-Banach type for Scott-continuous linear, sub- and superlinear functionals on continuous directed complete partially ordered cones, endowed with their Scott topologies, in analogy to the corresponding theorems for topological vector spaces in functional analysis (see Chapter 3). In the end, we show that our semantic domains work well for the language used by Morgan and McIver.

Polynomials on Riesz Spaces

Abstract In this paper we investigate polynomial mappings on Riesz spaces. We give a characterization of positivity of homogeneous polynomials in terms of forward differences. Finally we prove Hahn–Banach type extension theorems for positive and regular polynomial mappings.

Topological Cones: Foundations for a Domain Theoretical Semantics Combining Probability and Nondeterminism

There are more and more papers dealing with situations where probabilistic features occur together with ordinary nondeterminism. Based on the PhD thesis by R. Tix [R. Tix. Some results on Hahn-Banach type theorems for continuous d-cones. Theoretical Computer Science, 264:205–218, 2001], domain theoretical tools for combining probability with nondeterminism were developed in [R. Tix, K. Keimel, G.D. Plotkin. Semantic Domains Combining Probabilty and Nondeterminism. Electronic Notes in Theoretical Computer Science, 129:1–104, 2005]. A motivating situation was the semantics of an imperative language with both probabilistic and nondeterministic choice as considered by McIver and Morgan [A. McIver and C. Morgan. Partial correctness for probablistic demonic programs. Theoretical Computer Science, 266:513–541, 2001; A. McIver, and C. Morgan, Specification and Refinement of Probabilistic Systems. Monographs in Computer Science, Springer Verlag, 2004, 402 pages] for discrete state spaces. In [R. Tix, K. Keimel, G.D. Plotkin. Semantic Domains Combining Probabilty and Nondeterminism. Electronic Notes in Theoretical Computer Science, 129:1–104, 2005] discrete state spaces are replaced by arbitrary continuous domains. In this extended abstract we intend to show that the theory can be extended to larger classes of spaces including in particular stably locally compact state spaces. Proofs are mostly omitted. In dealing with domain theoretical versions of probabilities and spaces of probabilities (and, more generally, spaces of measures), domain theoretical variants of functional analytic concepts and tools like topological vector spaces, their topological duals and Hahn-Banach type separation theorems have to be developed.

A Characterization of Continuous Module Homomorphisms on Random Semi-Normed Modules and Its Applications

In this paper, a characterization of continuous module homomorphisms on random semi-normed modules is first given; then the characterization is further used to show that the Hahn-Banach type of extension theorem is still true for continuous module homomorphisms on random semi-normed modules.

ON HAHN–BANACH TYPE THEOREMS FOR HILBERT C*-MODULES

We establish three Hahn–Banach type extension criteria for (sets of) bounded C*-linear maps of Hilbert C*-modules to the underlying C*-algebras of coefficients. One criterion gives an alternative description of the property of C*-algebras to be monotone complete or additively complete.

Exact functionals and their core

Exact cooperative games or non-additive measures, coherent lower previsions and coherent risk measures are mathematically essentially the same. They all belong to the class of exact functionals on an arbitrary set of bounded functions. We investigate the exact functionals from a functional analytic point of view, i.e. we characterize this class by a norm, present a Hahn-Banach type theorem, provide a powerful construction method and adopt the concept of the core resp. σ-core from cooperative game theory.

Hahn-Banach Type Theorems for Locally Convex Cones

AbstractWe prove Hahn-Banach type theorems for linear functionals with values in R∪{+∞} on ordered cones, Using the concept of locally convex cones, we provide a sandwich theorem involving sub- and superlinear functionals which are allowed to attain infinite values. It render general versions of well-known extension and separation results. We describe the range of all linear functionals sandwiched between given sub- and superlinear functionals on an ordered cone. The results are of interest even in vector spaces, since we consider sublinear functionals that may attain the value +∞.