A note on the duality mapping of a locally uniformly convex Banach space

Let X be a real locally uniformly convex Banach space with normalized duality mapping J:X→2^X*. The purpose of this note is to show that for every R>0 and every x₀X there exists a function , which is nondecreasing and such that (r)>0 for r>0,(0)=0 and

for all . Simply, it is shown that the necessity part of the proof of the original analogous necessary and sufficient condition of Prüß, for real uniformly convex Banach spaces, goes over equally well in the present setting. This is a natural setting for the study of many existence problems in accretive and monotone operator theories.     

Every weakly compact set can be uniformly embedded into a reflexive Banach space

Based on an application of the Davis-Figiel-Johnson-Pelzyski procedure, this note shows that every weakly compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, we present the recent renorming theorems for reflexive spaces of Odell- Schlumprecht and Hajek-Johanis can be extended and localized to weakly compact convex subsets of an arbitrary Banach space.     

Functions operating from a complex Banach space to its real part

We consider functions operating from a complex Banach function space to its real part. We show among other things, that if for all in an ultraseparating Banach function space , then Re.

     

The AR-property for Roberts' example of a compact convex set with no extreme points Part 2: Application to the example

In this second part of our paper, we apply the result of Part 1 to show that the compact convex set with no extreme points, constructed by Roberts (1977), is an AR.

     

On Runge approximation in a Banach space

We show that on some open sets, more general than balls, Runge approximation is possible in certain Banach spaces, and also in certain complex Banach manifolds. We also show that there is an entire holomorphic curve in Hilbert space on which there is a bounded holomorphic function on the trace of a ball that has no bounded holomorphic extension to even a smaller concentric ball. Using the same technique we also prove that a form of Runge approximation better than an error function is not always possible.     

Properties of quadratic maps in a Banach space

Translated from Matematicheskie Zametki, Vol. 50, No. 4, pp. 10–20, October, 1991.     

Compact right multipliers on a Banach algebra related to locally compact semigroups

For a locally compact semigroup \({\mathcal{S}}\), let \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))\) be the Banach space of all μ-measurable (\(\mu\in M_{a}({\mathcal{S}})\)) functions vanishing at infinity, where \(M_{a}({\mathcal{S}})\) denotes the algebra of all measures in the measure algebra \(M({\mathcal{S}})\) of \({\mathcal{S}}\) with continuous translations. Here, we study right compact multipliers on the Banach algebra \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))^{*}\) equipped with an Arens product.     

The number of weakly compact sets which generate a Banach space

We consider the cardinal invariant CG(X) of the minimal number of weakly compact subsets which generate a Banach space X. We study the behavior of this index when passing to subspaces, its relation with the Lindelöf number in the weak topology and other related questions.     

Balayage of measures on a locally compact space

Analysis Mathematica - We develop a theory of inner balayage of a positive Radon measure μ of finite energy on a locally compact space X to arbitrary A ⊂ X, thereby generalizing...     

Arens multiplication on Banach algebras related to locally compact semigroups

Let S be a locally compact semigroup, let ω be a weight function on S, and let M_a (S, ω) be the weighted semigroup algebra of S. Let L^∞₀ (S;M_a (S, ω)) be the C^*‐algebra of allM_a (S, ω)‐measurable functions g on S such that g /ω vanishes at infinity. We introduce and study an Arens multiplication on L^∞₀ (S;M_a (S, ω))^* under which M_a (S, ω) is a closed ideal. We show that the weighted measure algebra M (S, ω) plays an important role in the structure of L^∞₀ (S;M_a (S, ω))^*. We then study Arens regularity of L^∞₀ (S;M_a (S, ω))^* and ist relation with Arens regularity of M_a (S, ω), M (S, ω) and the discrete convolution algebra ℓ¹(S, ω). As the main result, we prove that L^∞₀ (S;M_a (S, ω))^* is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The free Banach lattice generated by a Banach space

The free Banach lattice over a Banach space is introduced and analyzed. This generalizes the concept of free Banach lattice over a set of generators, and allows us to study the Nakano property and the density character of non-degenerate intervals on these spaces, answering some recent questions of B. de Pagter and A.W. Wickstead. Moreover, an example of a Banach lattice which is weakly compactly generated as a lattice but not as a Banach space is exhibited, thus answering a question of J. Diestel.     

The AR-property for Roberts' example of a compact convex set with no extreme points Part 1: general result

We prove that the original compact convex set with no extreme points, constructed by Roberts (1977) is an absolute retract, therefore is homeomorphic to the Hilbert cube. Our proof consists of two parts. In this first part, we give a sufficient condition for a Roberts space to be an AR. In the second part of the paper, we shall apply this to show that the example of Roberts is an AR.

     

The constructive theory of preference relations on a locally compact space

This paper, which is written within a rigorously constructive framework, deals with preference relations (strict weak orders) on a locally compact space X, and with the representation of such relations by continuous utility functions (order isomorphisms) from X into ℝ. Necessary conditions are given for finding the values of a utility function algorithmically in terms of the parameters when X is a locally compact, convex subset of R^N. These conditions single out the class of admissible preference relations, which are investigated in some detail. The paper concludes with some results on the algorithmic continuity of the process which assigns utility functions to admissible preference relations.The work of this paper can be regarded as a recursive development of preference and utility theory.     

Covering a Banach space

A well-known theorem by H. Corson states that if a Banach space admits a locally finite covering by bounded closed convex subsets, then it contains no infinite-dimensional reflexive subspace. We strengthen this result proving that if an infinite-dimensional Banach space admits a locally finite covering by bounded -closed subsets, then it is -saturated, thus answering a question posed by V. Klee concerning locally finite coverings of spaces. Moreover, we provide information about massiveness of the set of singular points in (PC) spaces.

     

Covering a compact set in a Banach space by an operator range of a Banach space with basis

A Banach space has the approximation property if and only if every compact set in is in the range of a one-to-one bounded linear operator from a space that has a Schauder basis. Characterizations are given for spaces and quotients of spaces in terms of covering compact sets in by operator ranges from spaces. A Banach space is a space if and only if every compact set in is contained in the closed convex symmetric hull of a basic sequence which converges to zero.

     

Semisimple Banach algebras generated by compact groups of operators on a Banach space

We prove that if τ is a strongly continuous representation of a compact group G on a Banach space X, then the weakly closed Banach algebra generated by the Fourier transforms with μ∈M(G) is a semisimple Banach algebra.     

Fixed point property for Banach algebras associated to locally compact groups

In this paper we investigate when various Banach algebras associated to a locally compact group G have the weak or weak^∗ fixed point property for left reversible semigroups. We proved, for example, that if G is a separable locally compact group with a compact neighborhood of the identity invariant under inner automorphisms, then the Fourier-Stieltjes algebra of G has the weak^∗ fixed point property for left reversible semigroups if and only if G is compact. This generalizes a classical result of T.C. Lim for the case when G is the circle group T.