105 research outputs found

    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

    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

    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

    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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