343 research outputs found

    The symmetric Radon-Nikod\'ym property for tensor norms

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    We introduce the symmetric-Radon-Nikod\'ym property (sRN property) for finitely generated s-tensor norms β\beta of order nn and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if β\beta is a projective s-tensor norm with the sRN property, then for every Asplund space EE, the canonical map ⊗~βn,sE′→(⊗~β′n,sE)′\widetilde{\otimes}_{\beta}^{n,s} E' \to \Big(\widetilde{\otimes}_{\beta'}^{n,s} E \Big)' is a metric surjection. This can be rephrased as the isometric isomorphism Qmin(E)=Q(E)\mathcal{Q}^{min}(E) = \mathcal{Q}(E) for certain polynomial ideal \Q. We also relate the sRN property of an s-tensor norm with the Asplund or Radon-Nikod\'{y}m properties of different tensor products. Similar results for full tensor products are also given. As an application, results concerning the ideal of nn-homogeneous extendible polynomials are obtained, as well as a new proof of the well known isometric isomorphism between nuclear and integral polynomials on Asplund spaces.Comment: 17 page

    Bounded holomorphic functions attaining their norms in the bidual

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    Under certain hypotheses on the Banach space XX, we prove that the set of analytic functions in Au(X)\mathcal{A}_u(X) (the algebra of all holomorphic and uniformly continuous functions in the ball of XX) whose Aron-Berner extensions attain their norms, is dense in Au(X)\mathcal{A}_u(X). The result holds also for functions with values in a dual space or in a Banach space with the so-called property (β)(\beta). For this, we establish first a Lindenstrauss type theorem for continuous polynomials. We also present some counterexamples for the Bishop-Phelps theorem in the analytic and polynomial cases where our results apply.Comment: Accepted in Publ. Res. Inst. Math. Sc

    Unconditionality in tensor products and ideals of polynomials, multilinear forms and operators

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    We study tensor norms that destroy unconditionality in the following sense: for every Banach space EE with unconditional basis, the nn-fold tensor product of EE (with the corresponding tensor norm) does not have unconditional basis. We establish an easy criterion to check weather a tensor norm destroys unconditionality or not. Using this test we get that all injective and projective tensor norms different from ε\varepsilon and π\pi destroy unconditionality, both in full and symmetric tensor products. We present applications to polynomial ideals: we show that many usual polynomial ideals never enjoy the Gordon-Lewis property. We also consider the unconditionality of the monomial basic sequence. Analogous problems for multilinear and operator ideals are addressed.Comment: 23 page

    Extending polynomials in maximal and minimal ideals

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    Given an homogeneous polynomial on a Banach space EE belonging to some maximal or minimal polynomial ideal, we consider its iterated extension to an ultrapower of EE 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 allow us to obtain symmetric versions of some basic results of the metric theory of tensor products.Comment: 13 page

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

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    We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral polynomials over a real Banach space XX is {±ϕk:ϕ∈X∗,∥ϕ∥=1}\{\pm \phi^k: \phi \in X^*, \| \phi\|=1\}. With this description we show that, for real Banach spaces XX and YY, if XX is a non trivial MM-ideal in YY, then ⨂^ϵk,sk,sX\hat\bigotimes^{k,s}_{\epsilon_{k,s}} X (the kk-th symmetric tensor product of XX endowed with the injective symmetric tensor norm) is \emph{never} an MM-ideal in ⨂^ϵk,sk,sY\hat\bigotimes^{k,s}_{\epsilon_{k,s}} Y. This result marks up a difference with the behavior of non-symmetric tensors since, when XX is an MM-ideal in YY, it is known that ⨂^ϵkkX\hat\bigotimes^k_{\epsilon_k} X (the kk-th tensor product of XX endowed with the injective tensor norm) is an MM-ideal in ⨂^ϵkkY\hat\bigotimes^k_{\epsilon_k} Y. Nevertheless, if XX is Asplund, we prove that every integral kk-homogeneous polynomial in XX has a unique extension to YY that preserves the integral norm. We explicitly describe this extension. We also give necessary and sufficient conditions (related with the continuity of the Aron-Berner extension morphism) for a fixed kk-homogeneous polynomial PP belonging to a maximal polynomial ideal \Q(^kX) to have a unique norm preserving extension to \Q(^kX^{**}). To this end, we study the relationship between the bidual of the symmetric tensor product of a Banach space and the symmetric tensor product of its bidual and show (in the presence of the BAP) that both spaces have `the same local structure'. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.Comment: 25 page

    A Lindenstrauss theorem for some classes of multilinear mappings

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    Under some natural hypotheses, we show that if a multilinear mapping belongs to some Banach multlinear ideal, then it can be approximated by multilinear mappings belonging to the same ideal whose Arens extensions simultaneously attain their norms. We also consider the class of symmetric multilinear mappings.Comment: 11 page

    Holomorphic Functions and polynomial ideals on Banach spaces

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    Given \u a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, H_{b\u}(E). We prove that, under very natural conditions verified by many usual classes of polynomials, the spectrum M_{b\u}(E) of this algebra "behaves" like the classical case of Mb(E)M_{b}(E) (the spectrum of Hb(E)H_b(E), the algebra of bounded type holomorphic functions). More precisely, we prove that M_{b\u}(E) can be endowed with a structure of Riemann domain over E"E" and that the extension of each f\in H_{b\u}(E) to the spectrum is an \u-holomorphic function of bounded type in each connected component. We also prove a Banach-Stone type theorem for these algebras.Comment: 19 page
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