31 research outputs found
Amalgamations of classes of Banach spaces with a monotone basis
It was proved by Argyros and Dodos that, for many classes of separable
Banach spaces which share some property , there exists an isomorphically
universal space that satisfies as well. We introduce a variant of their
amalgamation technique which provides an isometrically universal space in the
case that consists of spaces with a monotone Schauder basis. For example,
we prove that if is a set of separable Banach spaces which is analytic
with respect to the Effros-Borel structure and every is reflexive
and has a monotone Schauder basis, then there exists a separable reflexive
Banach space that is isometrically universal for
On binormality in non-separable Banach spaces
AbstractWe study binormality, a separation property of the norm and weak topologies of a Banach space. We show that every Banach space which belongs to a P-class is binormal. We also show that the asplundness of a Banach space is equivalent to a related separation property of its dual space
Orbit pseudometrics and a universality property of the Gromov-Hausdorff distance
We consider the notion of Borel reducibility between pseudometrics on
standard Borel spaces introduced and studied recently by C\'{u}th, Doucha and
Kurka, as well as the notion of an orbit pseudometric, a continuous version of
the notion of an orbit equivalence relation. It is well known that the relation
of isometry of Polish metric spaces is bireducible with a universal orbit
equivalence relation. We prove a version of this result for pseudometrics,
showing that the Gromov-Hausdorff distance of Polish metric spaces is
bireducible with a universal element in a certain class of orbit pseudometrics
Complexity of distances: Theory of generalized analytic equivalence relations
We generalize the notion of analytic/Borel equivalence relations, orbit
equivalence relations, and Borel reductions between them to their continuous
and quantitative counterparts: analytic/Borel pseudometrics, orbit
pseudometrics, and Borel reductions between them. We motivate these concepts on
examples and we set some basic general theory. We illustrate the new notion of
reduction by showing that the Gromov-Hausdorff distance maintains the same
complexity if it is defined on the class of all Polish metric spaces, spaces
bounded from below, from above, and from both below and above. Then we show
that is not reducible to equivalences induced by orbit pseudometrics,
generalizing the seminal result of Kechris and Louveau. We answer in negative a
question of Ben-Yaacov, Doucha, Nies, and Tsankov on whether balls in the
Gromov-Hausdorff and Kadets distances are Borel. In appendix, we provide new
methods using games showing that the distance-zero classes in certain
pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and
Tsankov.
There is a complementary paper of the authors where reductions between the
most common pseudometrics from functional analysis and metric geometry are
provided.Comment: Based on the feedback we received, we decided to split the original
version into two parts. The new version is now the first part of this spli
Large separated sets of unit vectors in Banach spaces of continuous functions
The paper concerns the problem whether a nonseparable \C(K) space must
contain a set of unit vectors whose cardinality equals to the density of
\C(K) such that the distances between every two distinct vectors are always
greater than one. We prove that this is the case if the density is at most
continuum and we prove that for several classes of \C(K) spaces (of arbitrary
density) it is even possible to find such a set which is -equilateral; that
is, the distance between every two distinct vectors is exactly 2.Comment: The second version does not contain new results, but it is
reorganized in order to distinguish our main contributions from what was
essentially know
Optimal quality of exceptional points for the Lebesgue density theorem
In spite of the Lebesgue density theorem, there is a positive such
that, for every non-trivial measurable set of real numbers, there is a
point at which both the lower densities of and of the complement of are
at least . The problem of determining the supremum of possible values
of this was studied in a paper of V. I. Kolyada, as well as in some
recent papers. We solve this problem in the present work.Comment: 45 page