The Geometry of Convex Transitive Banach Spaces

Throughout this paper, X will denote a Banach space, S=S(X)and B=B(X) will be the unit sphere and the closed unit ballof X, respectively, and G=G(X) will stand for the group of allsurjective linear isometries on X. Unless explicitly statedotherwise, all Banach spaces will be assumed to be real. Nevertheless,by passing to real structures, the results remain true for complexspaces. 1991 Mathematics Subject Classification 46B04, 46B10,46B22.     

On a Question of Haskell P. Rosenthal Concerning a Characterization of c0 and lp

The following property of a normalized basis in a Banach spaceis considered: any normalized block sequence of the basis hasa subsequence equivalent to the basis. Under uniformity or othernatural assumptions, a basis with this property is equivalentto the unit vector basis of c₀ or l_p. An analogous problem concerningspreading models is also addressed. 2000 Mathematics SubjectClassification 46B20 (primary), 46B15 (secondary).     

Continuity of Lie Isomorphisms of Banach Algebras

We prove that if A and B are semisimple Banach algebras, thenthe separating subspace of every Lie isomorphism from A ontoB is contained in the centre of B. 1991 Mathematics SubjectClassification 17B40, 17B60, 46H40.     

Banach Spaces Whose Algebras of Operators are Unitary: A Holomorphic Approach

An element u of a norm-unital Banach algebra A is said to beunitary if u is invertible in A and satisfies ||u|| = ||u^–1||= 1. The norm-unital Banach algebra A is called unitary if theconvex hull of the set of its unitary elements is norm-densein the closed unit ball of A. If X is a complex Hilbert space,then the algebra BL(X) of all bounded linear operators on Xis unitary by the Russo–Dye theorem. The question of whetherthis property characterizes complex Hilbert spaces among complexBanach spaces seems to be open. Some partial affirmative answersto this question are proved here. In particular, a complex Banachspace X is a Hilbert space if (and only if) BL(X) is unitaryand, for Y equal to X, X* or X** there exists a biholomorphicautomorphism of the open unit ball of Y that cannot be extendedto a surjective linear isometry on Y. 2000 Mathematics SubjectClassification 46B04, 46B10, 46B20.     

Arens Regularity of the Algebra of Operators on a Banach Space

A short proof is given that if E is a super-reflexive Banachspace, then B(E), the Banach algebra of operators on E withcomposition product, is Arens regular. Some remarks are madeon necessary conditions on E for B(E) to be Arens regular. 2000Mathematics Subject Classification 47L10 (primary), 46H05 (secondary).     

Products of Commuting Boolean Algebras of Projections and Banach Space Geometry

New criteria and Banach spaces are presented (for example, GL-spacesand Banach spaces with property () that ensure that the Booleanalgebra generated by a pair of bounded, commuting Boolean algebrasof projections is itself bounded. The notion of R-boundednessplays a fundamental role. It is shown that the strong operatorclosure of any R-bounded Boolean algebra of projections is necessarilyBade complete. Also, for a Dedekind -complete Banach latticeE, the Boolean algebra consisting of all band projections inE is R-bounded if and only if E has finite cotype. In this situation,every bounded Boolean algebra of projections in E is R-boundedand has a Bade complete strong closure. 2000 Mathematics SubjectClassification 46B20, 47L10 (primary), 46B42, 47B40, 47B60 (secondary).     

Interplay Between Distributional Estimates and Boundedness in Harmonic Analysis

In this paper, a proof is given that certain boundedness propertiesof operators yield distributional estimates that have exponentialdecay at infinity. Such distributional estimates imply localexponential integrability, and apply to many operators suchas m-linear Calderón–Zygmund operators and theirmaximal counterparts. 2000 Mathematics Subject Classification46B70, 42B99.     

Operators on Subspaces of Hereditarily Indecomposable Banach Spaces

We show that if X is a complex hereditarily indecomposable space,then every operator from a subspace Y of X to X is of the formI + S, where I is the inclusion map and S is strictly singular.1991 Mathematics Subject Classification 46B20, 47B99.     

Real Banach Spaces with Numerical Index 1

We show that an infinite-dimensional real Banach space withnumerical index 1 satisfying the Radon–Nikodym propertycontains l₁. It follows that a reflexive or quasi-reflexivereal Banach space cannot be re-normed to have numerical index1, unless it is finite-dimensional. 1991 Mathematics SubjectClassification 46B20.     

A Version of James' Theorem for Numerical Radius

We prove that if all the rank-one bounded operators on a Banachspace X attain their numerical radii, then X must be reflexive,but the converse does not hold. In fact, every reflexive spacewith basis can be renormed in such a way that there is a rank-oneoperator not attaining the numerical radius. 1991 MathematicsSubject Classification 47A12, 46B10.     

Factorization in finite-codimensional ideals of group algebras

Let G be a -compact, locally compact group and I be a closed2-sided ideal with finite codimension in L¹(G). It is shownthat there are a closed left ideal L having a right boundedapproximate identity and a closed right ideal R having a leftbounded approximate identity such that I = L + R. The proofuses ideas from the theory of boundaries of random walks ongroups. 2000 Mathematics Subject Classification: primary 43A20;secondary 42A85, 43A07, 46H10, 46H40, 60B11.     

Regularity of the Fractional Maximal Function

The purpose of this work is to show that the fractional maximaloperator has somewhat unexpected regularity properties. Themain result shows that the fractional maximal operator mapsL^p-spaces boundedly into certain first-order Sobolev spaces.It is also proved that the fractional maximal operator preservesfirst-order Sobolev spaces. This extends known results for theHardy-Littlewood maximal operator. 2000 Mathematics SubjectClassification 42B25, 46E35.     

Homogeneous Orthogonally Additive Polynomials on Banach Lattices

The main result in this paper is a representation theorem forhomogeneous orthogonally additive polynomials on Banach lattices.The representation theorem is used to study the linear spanof the set of zeros of homogeneous real-valued orthogonallyadditive polynomials. It is shown that in certain lattices everyelement can be represented as the sum of two or three zerosor, at least, can be approximated by such sums. It is also indicatedhow these results can be used to study weak topologies inducedby orthogonally additive polynomials on Banach lattices. 2000Mathematics Subject Classification 46G25, 46B42, 47B38.     

Lipschitz Images of Haar Null Sets

Let X be a separable infinite dimensional Banach space. Thereexist a closed set A X which contains a translate of any compactset in the unit ball of X, and a bi-Lipschitz homeomorphismf of X onto X so that every line in X intersects f(A) in a setof Lebesgue measure zero. 1991 Mathematics Subject Classification46B20.     

Ordered C*-Modules

In this first part of a study of ordered operator spaces, wedevelop the basic theory of ‘ordered C^*-bimodules’.A crucial role is played by ‘open tripotents’, aJB^*-triple variant of Akemann's notion of open projection. 2000Mathematics Subject Classification 46L08, 47L07 (primary), 46L07,47B60, 47L05 (secondary).     

Wild Spectral Behaviour of Anharmonic Oscillators

We show that the norms of the sequence of spectral projectionsof certain non-self-adjoint ordinary differential operatorscannot be polynomially bounded. The proof involves the investigationof ‘tame’ bi-orthogonal systems in Banach spaces.2000 Mathematics Subject Classification 34L10, 46B15.     

Locally uniformly rotund renorming and fragmentability

In this paper we characterize those Banach spaces E that admita locally uniformly rotund renorming by means of a linear topologicalcondition that is a particular case of the notion, introducedby J. E. Jayne, I. Namioka and C. A. Rogers, called countablecover by sets of small local diameter. This allows us to developa Decomposition Method and a Transfer Technique to obtain locallyuniformly rotund renormings. 1991 Mathematics Subject Classification:46B20.     

Tiling Infinite-Dimensional Normed Spaces

The present paper deals with real infinite-dimensional normedspaces; some of the main concepts here make sense, and havebeen treated in the literature, in the general context of topologicalHausdorff linear spaces over reals. A subset of a normed space X is a body if it is different fromX itself and is the closure of its non-empty interior. A coveringof X by bodies is called a tiling of X whenever any two differentmembers of it have disjoint interiors. The elements of sucha covering are called tiles. A tiling is bounded (respectivelyconvex) whenever each tile is bounded (respectively convex).1991 Mathematics Subject Classification 46B20.     

Embedding l{infty} into the Space of Bounded Operators on Certain Banach Spaces

Sufficient conditions are given on a Banach space X which ensurethat embeds in L(X), the space of all bounded linear operatorson X. A basic sequence e_n is said to be quasisubsymmetric iffor any two increasing sequences (k_n) and (l_n) of positive integerswith (k_n) (l_n) for all n, () dominates (). If a Banach space X has a seminormalized quasisubsymmetric basis then l embedsin l(X). 2000 Mathematics Subject Classification 46B28 (primary),46B03 (secondary).     

The Curvature and Index of Completely Positive Maps

We study conjugacy invariants for completely positive maps thatare inspired by the concept of curvature introduced for commutingd-tuples of contractions by Arveson. 2000 Mathematics SubjectClassification 46L53, 46L55, 46L57, 46L87, 47L55, 46L07, 46L08