105 research outputs found

### 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

### Extending polynomials in maximal and minimal ideals

Given an 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 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

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 $\mathcal{A}_u(X)$. The result holds also for
functions with values in a dual space or in a Banach space with the so-called
property $(\beta)$. 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

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

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