5,182 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 of order and prove a Lewis type
theorem for s-tensor norms with this property. As a consequence, if is
a projective s-tensor norm with the sRN property, then for every Asplund space
, the canonical map is a metric surjection. This
can be rephrased as the isometric isomorphism 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 -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 (the spectrum of , 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 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, -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
is . With this description we show
that, for real Banach spaces and , if is a non trivial -ideal in
, then (the -th symmetric
tensor product of endowed with the injective symmetric tensor norm) is
\emph{never} an -ideal in . This
result marks up a difference with the behavior of non-symmetric tensors since,
when is an -ideal in , it is known that
(the -th tensor product of endowed
with the injective tensor norm) is an -ideal in
. Nevertheless, if is Asplund, we prove
that every integral -homogeneous polynomial in has a unique extension to
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 -homogeneous polynomial
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 with unconditional basis, the -fold tensor
product of (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 and 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
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