Dual Banach Space Research Articles

Dentability and extreme points in Banach spaces

It is shown that a Banach space E has the Radon-Nikodym property (equivalently, every bounded subset of E is dentable) if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points. Using recent work of Namioka, some analogous results are obtained concerning weak ∗ strongly exposed points of weak ∗ compact convex subsets of certain dual Banach spaces.

On the Tensor Product of W ∗ Algebras

We develop the algebra underlying the of von Neumann in the language and spirit of Sakai's abstract W* algebras, and using the maximum spectrum of an abelian von Neumann algebra rather than a measure-theoretic surrogate. We are thus enabled to obtain the basic fact of the von Neumann as a special case of a weaker general decomposition theorem, valid without separability or type restrictions, and adapted to comparison with Wright's in the finite case. Introduction. The study of von Neumann algebras (weakly closed rings of operators on Hilbert space) has always been largely algebraic. Many of their properties follow from the spectral theorem, and can consequently be developed for the abstract AW* algebras of Kaplansky [6], [7]. However there are some more delicate consequences of weak closure which do not survive in a general A W* algebra. For questions of this kind we have the striking abstract characterization due to Sakai [9]: a C* algebra is* isomorphic to a von Neumann algebra if and only if it is a dual Banach space. This is taken as the definition of a W* algebra in [11], where much of the standard of von Neumann algebras is developed in a space-free manner, including the so-called reduction theory of von Neumann [16]. The central algebraic fact is found to be a representation of the W* tensor product Z A of an abelian W* algebra Z = L??(F, i) with another W* algebra A, as the algebra Z = LX (F, jt; A) of all essentially bounded weakly* measurable A-valued functions on F, with pointwise operations. This result is restricted to the case when A can be faithfully represented on a separable Hilbert space because of measure-theoretic difficulties. In this paper we prove an analogous theorem, without separability hypotheses, by working with the continuous Gelfand representation Z = C(2) of the abelian algebra, rather than a measure-theoretic one, the role of null sets being played by meager sets (i.e., sets of the first category) in 2, as follows: Theorem. Let Z = C(u) be an abelian W* algebra, and let A be any W* algebra, with predual A*, and let B = Z @ A be the W* tensor product. Let C* (2, A) be the Banach space of all w* continuous functions from 2 to A, with the supremum norm. Then there is a natural isometry

Bounded Mean Oscillation and Regulated Martingales

In the martingale context, the dual Banach space to ${H_1}$ is BMO in analogy with the result of Charles Fefferman [4] for the classical case. This theorem is an easy consequence of decomposition theorems for ${H_1}$-martingales which involve the notion of ${L_p}$-regulated ${L_1}$-martingales where $1 < p \leq \infty$. The strongest decomposition theorem is for $p = \infty$, and this provides full information about BMO. The weaker $p = 2$ decomposition is fundamental in the theory of martingale transforms.

Application of duality theory to a class of composite cost control problems

Let a control system be described by a continuous linear map ℒ* from the input spaceU* (some dual Banach space) into the output spaceX* (some finite-dimensional normed space). Within the class of control problems where the constraints and cost are expressed in terms of the norms on the input and output spaces, the following two have had extensive coverage: (i)minimum effort problem: find, from amongst all inputs which have corresponding outputs lying in some closed sphere inX* centered on some desired outputx d *, an output of minimum norm; and (ii)minimum deviation problem: find, from amongst all inputs lying in some closed sphere inU*, an input having corresponding output at a minimum distance fromx d *. However, thecomposite cost problem, where we seek to minimizeF(∥u*∥, ∥x d * −x*∥) over elements satisfyingx* = ℒ*u* (F a certain kind of convex functional), has not received the same attention. This paper presents results for the composite cost problem paralleling known results for the minimum effort and deviation problems. It is hoped that a gap in the literature is thereby filled. We show that (a) a solution exists, (b) the solution can be characterized in terms of some closed hyperplaneH inX, and (c)H can be computed as being an element on which some concave functional over closed hyperplanes inX achieves its maximum. The treatment allows of infinite-dimensional output spaces. We make extensive use of recently developed duality theory.

A Generalization to Dual Banach Spaces of a Theorem by Balakrishnan

A class of optimal control problems is studied in which the controls and outputs are taken as elements in Banach spaces, and the cost functions and constraints are expressible in terms of the norms on these spaces. The paper is principally concerned with generalizing certain results of Balakrishnan relating to optimal control in Hilbert space to this more general setting.

Bounded mean oscillation and regulated martingales

In the martingale context, the dual Banach space to H 1 {H_1} is BMO in analogy with the result of Charles Fefferman [4] for the classical case. This theorem is an easy consequence of decomposition theorems for H 1 {H_1} -martingales which involve the notion of L p {L_p} -regulated L 1 {L_1} -martingales where 1 &gt; p ≤ ∞ 1 &gt; p \leq \infty . The strongest decomposition theorem is for p = ∞ p = \infty , and this provides full information about BMO. The weaker p = 2 p = 2 decomposition is fundamental in the theory of martingale transforms.

H p -spaces of martingales, 0&lt;p?1

For 0 <p≦1 the dual Banach space to the space of Hp-martingales is characterized as a Lipschitz class of order α=l/p−1. When p=1 and α=0 this reduces to BMO being the dual of H1. In the course of the investigation a hierarchy of martingale spaces aLpis introduced with the property that the index p is in the range −∞ <1/p<∞. For l≦b≦q<∞ the space Lqis a subspace of bLq(equality when b=q) while for q<0 the space bLqcorresponds to a Lipschitz class of order α=−1/q.

M-structure in dual Banach spaces

A result previously known only for certain ordered Banach spaces is generalized to arbitrary real Banach spaces. Let ℒ be the Banach algebra of operators generated by theL-projections of a real Banach spaceU, and let ℳ (U * be the bounded operators on the dual spaceU * with adjoint in ℒ(U **. Then the adjoint operation maps ℒ (U) onto ℳ (U *). In particular, anyM-projection ofU * is weak* continuous.

Exposed faces of dual cones and peak-set criteria for function spaces

This paper gives conditions for a closed subcone of a weak ∗ closed cone in a Banach dual space to be exposed by a weak ∗ continuous linear functional. This set-up is applied to the study of complex-valued and vector-valued continuous function spaces on a compact Hausdorff space to deduce peak-set criteria. In the case of complex-valued function spaces peak-set conditions are given in terms of gages on certain cones of complex measures. These conditions are shown to imply various known peak-set criteria involving annihilating measures.

The states of a Banach algebra generate the dual

In this paper we prove that the states of a unital Banach algebra generate the dual Banach space as a linear space (Theorem 2). This is a result of R. T. Moore (4, Theorem 1(a)) who uses a decomposition of measures in his proof. In the proof given here the measure theory is replaced by a Hahn-Banach separation argument. We shall let A denote a unital Banach algebra over the complex field, and D(1) denote {f ∈ A′: ‖f‖ = f(1) = 1} where A′ is the dual of A. The motivation of Moore's results is the theorem that in a C*-algebra every continuous linear functional is a linear combination of four states (the states are the elements of D(1)) (see (2, 2.6.4, 2.1.9, 1.1.10)).

Barycenters of Measures on Certain Noncompact Convex Sets

Each norm closed and bounded convex subset K of a separable dual Banach space is, according to a theorem of Bessaga and Pełczynski, the norm closed convex hull of its extreme points. It is natural to expect that this theorem may be reformulated as an integral representation theorem, and in this connection we have examined the extent to which the Choquet theory applies to such sets. Two integral representation theorems are proved and an example is included which shows that a sharp result obtains for certain noncompact sets. In addition, the set of extreme points of K is shown to be $\mu$-measurable for each finite regular Borel measure $\mu$, hence eliminating certain possible measure-theoretic difficulties in proving a general integral representation theorem. The last section is devoted to the study of a class of extreme points (called pinnacle points) which share important geometric properties with extreme points of compact convex sets in locally convex spaces. A uniqueness result is included for certain simplexes all of whose extreme points are pinnacle points.

Barycenters of measures on certain noncompact convex sets

Each norm closed and bounded convex subset K of a separable dual Banach space is, according to a theorem of Bessaga and Pelczynski, the norm closed convex hull of its extreme points. It is natural to expect that this theorem may be reformulated as an integral representation theorem, and in this connection we have examined the extent to which the Choquet theory applies to such sets. Two integral representation theorems are proved and an example is included which shows that a sharp result obtains for certain noncompact sets. In addition, the set of extreme points of K is shown to be a-measurable for each finite regular Borel measure , hence eliminating certain possible measure-theoretic difficulties in proving a general integral representation theorem. The last section is devoted to the study of a class of extreme points (called pinnacle points) which share important geometric properties with extreme points of compact convex sets in locally convex spaces. A uniqueness result is included for certain simplexes all of whose extreme points are pinnacle points. Introduction. In 1966 Bessaga and Pelczynski [1] proved that each norm closed and bounded convex subset of a separable dual Banach space is the norm closed convex hull of its extreme points, thus providing a type theorem for a certain collection of noncompact sets. (A more elementary proof of this result has been given in [11].) For compact convex sets (in locally convex spaces) the Krein-Milman theorem can easily be reformulated as an integral representation theorem, using measures which are supported by the closure of the set of extreme points (see for example [13, p. 6]). This formulation has a large number of applications to analysis, probability, etcetera. A much more precise kind of representation theorem is provided by Choquet theory, where the measures (at least in the metrizable case) are supported by the set of extreme points, not just the closure. (See [5] and [2]. For a comprehensive and readable account see [13].) The BessagaPelczynski theorem would probably find greater applicability if it too could be reformulated as an integral representation theorem. As a step towards solving this problem we investigate to what extent the Choquet theory applies to the sets described in the Bessaga-Pelczynski theorem. We take this opportunity to thank Professor R. R. Phelps for directing the author's thesis from which this paper is Received by the editors October 17, 1969. AMS 1970 subject classifications. Primary 46B99; Secondary 46G99, 28A40.

A note on simplex spaces

Let X be a real Banach space which is partially ordered by a closed cone C. Let X* denote the Banach dual space, and letX is called a simplex space (2) if (X*, C*,‖·‖) is an AL-space. We proved in (3) that X is a simplex space if and only if X is an approximate order-unit-normed space and has the Riesz decomposition property. When the given norm on X is an order-unit norm, some representation theorems are obtained by Effros ((2), theorem 4·8) and Sterrmer ((6), theorem 4·3). In this note, we shall study the representation problem for more general simplex spaces. One of our aims is to prove that X is equivalent to C0(ω) if and only if X has the Urysohn property (for definitions, see below).

On strong measurability of Banach valued functions

1. Statement of the Theorem. Let (Q, a, A) be a measure space, , being a finite positive measure on the o-field a of subsets of U. Let B be a Banach space, and S a set of convex equilibrated weakly compact subsets of B. A mapping f from Q into B is said to be strongly ,measurable if it takes its values in a separable subset of B and, for every ball B in B, ft'(B) belongs to the u-completion of a. We denote by B' the Banach dual space of B. Then we have the following

On the Homeomorphic Affine Embedding of a Locally Compact Cone into a Banach Dual Space Endowed with the Vague Topology

Proceedings of the London Mathematical SocietyVolume s3-14, Issue 3 p. 399-414 Articles On the Homeomorphic Affine Embedding of a Locally Compact Cone into a Banach Dual Space Endowed with the Vague Topology D. A. Edwards, D. A. Edwards Mathematical Institute, 10 Parks Road, OxfordSearch for more papers by this author D. A. Edwards, D. A. Edwards Mathematical Institute, 10 Parks Road, OxfordSearch for more papers by this author First published: July 1964 https://doi.org/10.1112/plms/s3-14.3.399Citations: 33AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes3-14, Issue3July 1964Pages 399-414 RelatedInformation