70 research outputs found
Quiz your maths: do the uniformly continuous functions on the line form a ring?
The paper deals with the interplay between boundedness, order and ring
structures in function lattices on the line and related metric spaces. It is
shown that the lattice of all Lipschitz functions on a normed space is
isomorphic to its sublattice of bounded functions if and only if has
dimension one. The lattice of Lipschitz functions on carries a "hidden"
-ring structure with a unit, and the same happens to the (larger) lattice of
all uniformly continuous functions for a wide variety of metric spaces.
An example of a metric space whose lattice of uniformly continuous functions
supports no unital -ring structure is provided.Comment: 14 pages, to be published in Proceedings of the American Mathematical
Societ
Fine structure of the homomorphisms of the lattice of uniformly continuous functions on the line
We provide a representation of the homomorphisms , where is the lattice of all uniformly continuous on the line. The
resulting picture is sharp enough to describe the fine topological structure of
the space of such homomorphisms.Comment: 11 pages, 1 figur
Nonlinear Centralizers in Homology II. The Schatten classes
The paper computes the spaces of extensions for the Schatten classes when
they are regarded in its natural module structure over the algebra of bounded
operators on the ground Hilbert space.Comment: 30 page
An example regarding Kalton's paper "Isomorphisms between spaces of vector-valued continuous functions"
The paper alluded to in the title contains the following striking result: Let
be the unit interval and the Cantor set. If is a quasi Banach
space containing no copy of which is isomorphic to a closed subspace of a
space with a basis and is linearly homeomorphic to ,
then is locally convex, i.e., a Banach space.
It is shown that Kalton result is sharp by exhibiting non locally convex
quasi Banach spaces X with a basis for which and are
isomorphic. Our examples are rather specific and actually in all cases X is
isomorphic to if is a metric compactum of finite covering
dimension.Comment: 4 page
Stability constants and the homology of quasi-Banach spaces
We affirmatively solve the main problems posed by Laczkovich and Paulin in
\emph{Stability constants in linear spaces}, Constructive Approximation 34
(2011) 89--106 (do there exist cases in which the second Whitney constant is
finite while the approximation constant is infinite?) and by Cabello and
Castillo in \emph{The long homology sequence for quasi-Banach spaces, with
applications}, Positivity 8 (2004) 379--394 (do there exist Banach spaces
for which \Ext(X,Y) is Hausdorff and non-zero?). In fact, we show that these
two problems are the same.Comment: This paper is to appear in Israel Journal of Mathematic
On the bounded approximation property on subspaces of when and related issues
This paper studies the bounded approximation property (BAP) in quasi Banach
spaces. In the first part of the paper we show that the kernel of any
surjective operator has the BAP when has it and ,
which is an analogue of the corresponding result of Lusky for Banach spaces. We
then obtain and study nonlocally convex versions of the Kadec-Pe\l
czy\'nski-Wojtaszczyk complementably universal spaces for Banach spaces with
the BAP
Complex interpolation and twisted twisted Hilbert spaces
We show that Rochberg's generalizared interpolation spaces
arising from analytic families of Banach spaces form exact sequences . We study
some structural properties of those sequences; in particular, we show that
nontriviality, having strictly singular quotient map, or having strictly
cosingular embedding depend only on the basic case . If we focus on the
case of Hilbert spaces obtained from the interpolation scale of
spaces, then becomes the well-known Kalton-Peck space;
we then show that is (or embeds in, or is a quotient of) a
twisted Hilbert space only if , which solves a problem posed by David
Yost; and that it does not contain complemented unless . We
construct another nontrivial twisted sum of with itself that contains
complemented
On the category of quotient Banach spaces after Wegner
We study Waelbroeck's category of Banach quotients after Wegner, focusing on
its basic homological and functional analytic properties
Twisting non-commutative spaces
The paper makes the first steps into the study of extensions ("twisted sums")
of noncommutative -spaces regarded as Banach modules over the underlying
von Neumann algebra .
Our approach combines Kalton's description of extensions by centralizers
(these are certain maps which are, in general, neither linear nor bounded) with
a general principle, due to Rochberg and Weiss saying that whenever one finds a
given Banach space as an intermediate space in a (complex) interpolation
scale, one automatically gets a self-extension
For semifinite algebras, considering as an
interpolation space between and its predual one
arrives at a certain self-extension of that is a kind of noncommutative
Kalton-Peck space and carries a natural bimodule structure. Some interesting
properties of these spaces are presented.
For general algebras, including those of type III, the interpolation
mechanism produces two (rather than one) extensions of one sided modules, one
of left-modules and the other of right-modules. Whether or not one can find
(nontrivial) self-extensions of bimodules in all cases is left open
On Mazur rotations problem and its multidimensional versions
The article is a survey related to a classical unsolved problem in Banach
space theory, appearing in Banach's famous book in 1932, and known as the Mazur
rotations problem. Although the problem seems very difficult and rather
abstract, its study sheds new light on the importance of norm symmetries of a
Banach space, demonstrating sometimes unexpected connections with renorming
theory and differentiability in functional analysis, with topological group
theory and the theory of representations, with the area of amenability, with
Fra\"iss\'e theory and Ramsey theory, and led to development of concepts of
interest independent of Mazur problem. This survey focuses on results that have
been published after 2000, stressing two lines of research which were developed
in the last ten years. The first one is the study of approximate versions of
Mazur rotations problem in its various aspects, most specifically in the case
of the Lebesgue spaces Lp. The second one concerns recent developments of
multidimensional formulations of Mazur rotations problem and associated
results. Some new results are also included.Comment: 57 pages. This survey will be published in the special issue of the
S\~ao Paulo Journal of Mathematical Sciences dedicated to the Golden Jubilee
of the Institute of Mathematics and Statistics of the University of S\~ao
Paul
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