Aron-Berner Extension Research Articles

Grothendieck-type characterization of representable multilinear mappings

Let μ be a finite measure, T : L 1 ( μ ) → X be an operator into a Banach space X, and I : L ∞ ( μ ) → L 1 ( μ ) be the natural inclusion. A well-known theorem due to Grothendieck states that T is representable if and only T ∘ I is nuclear. In a 2022 paper we gave a multilinear version of this theorem replacing nuclearity by the weaker property that we call the left integral ℓ 1 -factorization property. Here we prove the rather surprising result that the multilinear version is also true for nuclearity. As an application, we show that the composition P ∘ T of an integral operator T with a polynomial P whose Aron–Berner extension to the bidual takes values in the original range space is nuclear. This extends (and strengthens) to the polynomial setting a result on composition of operators also due to Grothendieck.

Aron–Berner extensions of almost Dunford–Pettis multilinear operators

Aron–Berner extensions of almost Dunford–Pettis multilinear operators

A note on numerical radius attaining mappings

We prove that if every bounded linear operator (or N N -homogeneous polynomials) on a Banach space X X with the compact approximation property attains its numerical radius, then X X is a finite dimensional space. Moreover, we present an improvement of the polynomial James’ theorem for numerical radius proved by Acosta, Becerra Guerrero and Galán [Q. J. Math. 54 (2003), pp. 1–10]. Finally, the denseness of weakly (uniformly) continuous 2 2 -homogeneous polynomials on a Banach space whose Aron-Berner extensions attain their numerical radii is obtained.

Bidual extensions of Riesz multimorphisms

We prove that all Arens extensions of finite rank Riesz multimorphisms taking values in Archimedean Riesz spaces coincide and are Riesz multimorphisms. We also prove that, for a class of Banach lattices F, which includes F=c0,ℓp,c0(ℓp),ℓp(c0), ℓp(ℓs),1<p,s<∞, among many others, all Aron-Berner extensions of any F-valued Riesz multimorphism between Banach lattices are Riesz multimorphisms.

Representable and Radon-Nikodým polynomials

Every k-homogeneous (continuous) polynomial between Banach spaces admits a unique Aron-Berner extension to the biduals. Our main result states that, for every σ-finite measure µ, every polynomial P ∈ with Y -valued Aron-Berner extension is representable, in other words, it factors through ℓ 1 in the form P = A◦Q where Q is a polynomial and A is an operator. This may be viewed as a polynomial strengthening of the Dunford-Pettis-Phillips theorem stating that every weakly compact operator on L 1(µ) is representable. We introduce the Radon-Nikodým polynomials and show that every polynomial with Y -valued Aron-Berner extension is Radon-Nikodým. Finally, we prove that every Radon-Nikodým polynomial is unconditionally converging.

Synnatzschke’s theorem for polynomials

We establish a multilinear generalisation of Synnatzschke’s Theorem for regular operators on Banach lattices, with the Arens adjoint taking the place of the transpose. Using this result, we show that the Aron–Berner extension of a regular homogeneous polynomial to the order continuous bidual preserves the absolute value. As a consequence, it follows that the regular norm is preserved by the Aron–Berner extension.

Aron–Berner extensions of triple maps with application to the bidual of Jordan Banach triple systems

By extending the notion of Arens regularity of bilinear mappings, we say that a bounded trilinear map on the Cartesian product of Banach spaces is Aron–Berner regular when all its six Aron–Berner extensions to the Cartesian product of the bidual spaces coincide. We give some results on the Aron–Berner regularity of certain trilinear maps. We then focus on the bidual, E⁎⁎, of a Jordan Banach triple system (E,π), and investigate those conditions under which E⁎⁎ is itself a Jordan Banach triple system under each of the Aron–Berner extensions of the triple product π. We also compare these six triple products with those arising from certain ultrafilters based on the ultrapower formulation of the principle of local reflexivity. In particular, we examine the Aron–Berner triple products on the bidual of a JB⁎-triple in relation with the so-called Dineen's theorem. Some illuminating examples are included and some questions are also left undecided.

Some progress on the polynomial Dunford–Pettis property

A Banach space E has the Dunford-Pettis property ( DPP , for short) if every weakly compact (linear) operator on E is completely continuous. In 1979 R. A. Ryan proved that E has the DPP if and only if every weakly compact polynomial on E is completely continuous. Every k -homogeneous (continuous) polynomial P\in\mathcal P(^k\!E,F) between Banach spaces E and F admits an extension \widetilde P\in\mathcal P(^k\!E^{**},F^{**}) to the biduals called the Aron–Berner extension. The Aron–Berner extension of every weakly compact polynomial P\in{\mathcal P}(^k\!E,F) is F -valued, that is, \widetilde P(E^{**})\subseteq F , but there are non-weakly compact polynomials with F -valued Aron–Berner extension. For Banach spaces F with weak-star sequentially compact dual unit ball B_{F^*} , we strengthen Ryan's result by showing that E has the DPP if and only if every polynomial P\in\mathcal P(^k\!E,F) with F -valued Aron–Berner extension is completely continuous. This gives a partial answer to a question raised in 2003 by I. Villanueva and the second named author. They proved the result for spaces E such that every operator from E into its dual E^* is weakly compact, but the question remained open for other spaces.

Bounded Holomorphic Functions Attaining their Norms in the Bidual

Under certain hypotheses on the Banach space X , we prove that the set of analytic functions in \mathcal A_u (X) (the algebra of all holomorphic and uniformly continuous functions in the ball of X ) whose Aron–Berner extensions attain their norms is dense in \mathcal A_u (X) . This Lindenstrauss type result also holds for functions with values in a dual space or in a Banach space with the so-called property ( \beta ). We show that the Bishop–Phelps theorem does not hold for \mathcal A_u (c_0, Z'') for a certain Banach space Z , while our Lindenstrauss theorem does. In order to obtain our results, we first handle their polynomial cases.

On geometric extension of polynomials on Banach spaces

We consider some questions related to Aron-Berner extensions of polynomials on infinitely dimensional complex Banach spaces, using natural extensions of their zeros.

The Dineen problem for homogeneous orthogonally-additive polynomials

In this paper, we study the extensions to the Dedekind completion and the Aron-Berner extensions of orthogonally-additive polynomials and we show that this class is preserved by these extensions, which gives a positive answer to a problem posed by Dineen (LNM 332:77–111, 1973).

On the polynomial Lindenstrauss theorem

Under certain hypotheses on the Banach space X, we show that the set of N-homogeneous polynomials from X to any dual space, whose Aron–Berner extensions are norm attaining, is dense in the space of all continuous N-homogeneous polynomials. To this end we prove an integral formula for the duality between tensor products and polynomials. We also exhibit examples of Lorentz sequence spaces for which there is no polynomial Bishop–Phelps theorem, but our results apply. Finally we address quantitative versions, in the sense of Bollobás, of these results.

Nicodemi sequences of operators between spaces of multilinear mappings

We obtain results on three aspects of Nicodemi extensions of multilinear mappings between Banach spaces: (i) subspace invariance, (ii) the norms of the extension operators, (iii) when Aron–Berner extensions are Nicodemi extensions.

Geometry of integral polynomials, M-ideals and unique norm preserving extensions

We use the Aron–Berner extension to prove that the set of extreme points of the unit ball of the space of integral k-homogeneous polynomials over a real Banach space X is { ± ϕ k : ϕ ∈ X ⁎ , ‖ ϕ ‖ = 1 } . With this description we show that, for real Banach spaces X and Y, if X is a nontrivial M-ideal in Y, then ⊗ ˆ ε k , s k , s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in ⊗ ˆ ε k , s k , s Y . This result marks up a difference with the behavior of nonsymmetric tensors since, when X is an M-ideal in Y, it is known that ⊗ ˆ ε k k X (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in ⊗ ˆ ε k k Y . Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.

Local complementation and the extension of bilinear mappings

AbstractWe study different aspects of the connections between local theory of Banach spaces and the problem of the extension of bilinear forms from subspaces of Banach spaces. Among other results, we prove that if X is not a Hilbert space then one may find a subspace of X for which there is no Aron–Berner extension. We also obtain that the extension of bilinear forms from all the subspaces of a given X forces such X to contain no uniform copies of ℓpn for p ∈ [1, 2). In particular, X must have type 2 − ϵ for every ϵ &gt; 0. Also, we show that the bilinear version of the Lindenstrauss–Pełczyński and Johnson–Zippin theorems fail. We will then consider the notion of locally α-complemented subspace for a reasonable tensor norm α, and study the connections between α-local complementation and the extendability of α*-integral operators.

Quasi-completely continuous multilinear operators

We introduce the concept of quasi-completely continuous multilinear operators and use this concept to characterize, for a wide class of Banach spaces X1, …, Xk, the multilinear operators T : X1 × … × Xk → X with an X-valued Aron–Berner extension.

Extending Polynomials in Maximal and Minimal Ideals

Given a homogeneous polynomial on a Banach space E belonging to some maximal or minimal polynomial ideal, we consider its iterated extension to an ultrapower of E and prove that this extension remains in the ideal and has the same ideal norm. As a consequence, we show that the Aron–Berner extension is a well defined isometry for any maximal or minimal ideal of homogeneous polynomials. This allows us to obtain symmetric versions of some basic results of the metric theory of tensor products.

Composition, numerical range and Aron-Berner extension

Given an entire mapping $f\in \mathcal{H}_b(X,X)$ of bounded type from a Banach space $X$ into $X$, we denote by $\overline{f}$ the Aron-Berner extension of $f$ to the bidual $X^{\ast\ast}$ of $X$. We show that $\overline{g\circ f} = \overline{g}\circ \overline{f}$ for all $f, g\in \mathcal{H}_b(X,X)$ if $X$ is symmetrically regular. We also give a counterexample on $l_1$ such that the equality does not hold. We prove that the closure of the numerical range of $f$ is the same as that of $\bar{f}$.

Orthogonally Additive Polynomials over C(K) are Measures—A Short Proof

We give a simple proof of the fact that orthogonally additive polynomials on C(K) are represented by regular Borel measures over K. We also prove that the Aron-Berner extension preserves this class of polynomials.

Extension of vector-valued integral polynomials

We study the extendibility of integral vector-valued polynomials on Banach spaces. We prove that an X-valued Pietsch-integral polynomial on E extends to an X-valued Pietsch-integral polynomial on any space F containing E, with the same integral norm. This is not the case for Grothendieck-integral polynomials: they do not always extend to X-valued Grothendieck-integral polynomials. However, they are extendible to X-valued polynomials. The Aron–Berner extension of an integral polynomial is also studied. A canonical integral representation is given for domains not containing ℓ 1 .