343 research outputs found

### 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

### 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

### 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

### 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 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

### A Lindenstrauss theorem for some classes of multilinear mappings

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

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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