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

We introduce the symmetric-Radon-Nikod\'ym property (sRN property) for finitely generated s-tensor norms $\beta$ of order $n$ and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if $\beta$ is a projective s-tensor norm with the sRN property, then for every Asplund space $E$, the canonical map $\widetilde{\otimes}_{\beta}^{n,s} E' \to \Big(\widetilde{\otimes}_{\beta'}^{n,s} E \Big)'$ is a metric surjection. This can be rephrased as the isometric isomorphism $\mathcal{Q}^{min}(E) = \mathcal{Q}(E)$ 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 $n$-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

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 $M_{b}(E)$ (the spectrum of $H_b(E)$, 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 $E"$ 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

Geometry of integral polynomials, MM-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 polynomials over a real Banach space $X$ is $\{\pm \phi^k: \phi \in X^*, \| \phi\|=1\}$. With this description we show that, for real Banach spaces $X$ and $Y$, if $X$ is a non trivial $M$-ideal in $Y$, then $\hat\bigotimes^{k,s}_{\epsilon_{k,s}} X$ (the $k$-th symmetric tensor product of $X$ endowed with the injective symmetric tensor norm) is \emph{never} an $M$-ideal in $\hat\bigotimes^{k,s}_{\epsilon_{k,s}} Y$. This result marks up a difference with the behavior of non-symmetric tensors since, when $X$ is an $M$-ideal in $Y$, it is known that $\hat\bigotimes^k_{\epsilon_k} X$ (the $k$-th tensor product of $X$ endowed with the injective tensor norm) is an $M$-ideal in $\hat\bigotimes^k_{\epsilon_k} Y$. Nevertheless, if $X$ is Asplund, we prove that every integral $k$-homogeneous polynomial in $X$ has a unique extension to $Y$ 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 $k$-homogeneous polynomial $P$ 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

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

We study tensor norms that destroy unconditionality in the following sense: for every Banach space $E$ with unconditional basis, the $n$-fold tensor product of $E$ (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 $\varepsilon$ and $\pi$ 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