178 research outputs found
Vector-valued invariant means revisited once again
Banach spaces that are complemented in the second dual are characterised
precisely as those spaces which enjoy the property that for every amenable
semigroup there exists an -valued analogue of an invariant mean defined
on the Banach space of all bounded -valued functions on . This was first
observed by Bustos Domecq (J. Math. Anal. Appl., 2002), however the original
proof was slightly flawed as remarked by Lipecki. The primary aim of this note
is to present a corrected version of the proof. We also demonstrate that
universally separably injective spaces always admit invariant means with
respect to countable amenable semigroups, thus such semigroups are not rich
enough to capture complementation in the second dual as spaces falling into
this class need not be complemented in the second dual
Uncountable sets of unit vectors that are separated by more than 1
Let be a Banach space. We study the circumstances under which there
exists an uncountable set of unit vectors such that
for distinct . We prove that such a set exists
if is quasi-reflexive and non-separable; if is additionally
super-reflexive then one can have for some
that depends only on . If is a non-metrisable compact,
Hausdorff space, then the unit sphere of also contains such a subset;
if moreover is perfectly normal, then one can find such a set with
cardinality equal to the density of ; this solves a problem left open by S.
K. Mercourakis and G. Vassiliadis.Comment: to appear in Studia Mat
A short proof of the fact that the matrix trace is the expectation of the numerical values
Using the fact that the normalised matrix trace is the unique linear
functional on the algebra of matrices which satisfies
and for all matrices and , we derive a
well-known formula expressing the normalised trace of a complex matrix as
the expectation of the numerical values of ; that is the function , where ranges the unit sphere of
The ideal of weakly compactly generated operators acting on a Banach space
We call a bounded linear operator acting between Banach spaces weakly
compactly generated ( for short) if its range is contained in a
weakly compactly generated subspace of its codomain. This notion simultaneously
generalises being weakly compact and having separable range. In a comprehensive
study of the class of operators, we prove that it forms a closed
surjective operator ideal and investigate its relations to other classical
operator ideals. By considering the th long James space
, we show how properties of the ideal of
operators (such as being the unique maximal ideal) may be used
to derive results outside ideal theory. For instance, we identify the
-group of as the additive group of
integers
A reflexive Banach space whose algebra of operators is not a Grothendieck space
By a result of Johnson, the Banach space contains a complemented copy of . We identify
with a complemented subspace of the space of (bounded, linear) operators on
the reflexive space (, thus giving a negative answer to the problem posed in the
monograph of Diestel and Uhl which asks whether the space of operators on a
reflexive Banach space is Grothendieck
Restricting uniformly open surjections
We employ the theory of elementary submodels to improve a recent result by
Aron, Jaramillo and Le Donne (Ann. Acad. Sci. Fenn. Math., to appear)
concerning restricting uniformly open, continuous surjections to smaller
subspaces where they remain surjective. To wit, suppose that and are
metric spaces and let be a continuous surjection. If is
complete and is uniformly open, then contains a~closed subspace
with the same density as such that restricted to is still uniformly
open and surjective. Moreover, if is a Banach space, then may be taken
to be a closed linear subspace. A counterpart of this theorem for uniform
spaces is also established.Comment: 5 p
When is multiplication in a Banach algebra open?
We develop the theory of Banach algebras whose multiplication (regarded as a
bilinear map) is open. We demonstrate that such algebras must have topological
stable rank 1, however the latter condition is strictly weaker and implies only
that products of non-empty open sets have non-empty interior. We then
investigate openness of convolution in semigroup algebras resolving in the
negative a problem of whether convolution in is open. By
appealing to ultraproduct techniques, we demonstrate that neither in
nor in convolution is uniformly open.
The problem of openness of multiplication in Banach algebras of bounded
operators on Banach spaces and their Calkin algebras is also discussed.Comment: 15 p
Large cardinals and continuity of coordinate functionals of filter bases in Banach spaces
Assuming the existence of certain large cardinal numbers, we prove that for
every projective filter over the set of natural numbers,
-bases in Banach spaces have continuous coordinate functionals. In
particular, this applies to the filter of statistical convergence, thereby we
solve a problem by V. Kadets (at least under the presence of certain large
cardinals). In this setting, we recover also a result of Kochanek who proved
continuity of coordinate functionals for countably generated filters (Studia
Math., 2012).Comment: 10 p
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