164,277 research outputs found
Local Operator Multipliers and Positivity
We establish an unbounded version of Stinespring's Theorem and a lifting
result for Stinespring representations of completely positive modular maps
defined on the space of all compact operators. We apply these results to study
positivity for Schur multipliers. We characterise positive local Schur
multipliers, and provide a description of positive local Schur multipliers of
Toeplitz type. We introduce local operator multipliers as a non-commutative
analogue of local Schur multipliers, and obtain a characterisation that extends
earlier results concerning operator multipliers and local Schur multipliers. We
provide a description of the positive local operator multipliers in terms of
approximation by elements of canonical positive cones.Comment: 31 page
Unconditionality, Fourier multipliers and Schur multipliers
Let be an infinite locally compact abelian group. If is Banach space,
we show that if every bounded Fourier multiplier on has the
property that T\ot Id_X is bounded on then the Banach space is
isomorphic to a Hilbert space. Moreover, if , , we prove
that there exists a bounded Fourier multiplier on which is not
completely bounded. Finally, we examine unconditionality from the point of view
of Schur multipliers. More precisely, we give several necessary and sufficient
conditions to determine if an operator space is completely isomorphic to an
operator Hilbert space.Comment: minor corrections; 17 pages ; to appear in Colloquium Mathematicu
Schur and operator multipliers
Schur multipliers were introduced by Schur in the early 20th century and have
since then found a considerable number of applications in Analysis and enjoyed
an intensive development. Apart from the beauty of the subject in itself,
sources of interest in them were connections with Perturbation Theory, Harmonic
Analysis, the Theory of Operator Integrals and others. Advances in the
quantisation of Schur multipliers were recently made by Kissin and Shulman. The
aim of the present article is to summarise a part of the ideas and results in
the theory of Schur and operator multipliers. We start with the classical Schur
multipliers defined by Schur and their characterisation by Grothendieck, and
make our way through measurable multipliers studied by Peller and Spronk,
operator multipliers defined by Kissin and Shulman and, finally,
multidimensional Schur and operator multipliers developed by Juschenko and the
authors. We point out connections of the area with Harmonic Analysis and the
Theory of Operator Integrals
Multidimensional operator multipliers
We introduce multidimensional Schur multipliers and characterise them
generalising well known results by Grothendieck and Peller. We define a
multidimensional version of the two dimensional operator multipliers studied
recently by Kissin and Shulman. The multidimensional operator multipliers are
defined as elements of the minimal tensor product of several C*-algebras
satisfying certain boundedness conditions. In the case of commutative
C*-algebras, the multidimensional operator multipliers reduce to continuous
multidimensional Schur multipliers. We show that the multipliers with respect
to some given representations of the corresponding C*-algebras do not change if
the representations are replaced by approximately equivalent ones. We establish
a non-commutative and multidimensional version of the characterisations by
Grothendieck and Peller which shows that universal operator multipliers can be
obtained as certain weak limits of elements of the algebraic tensor product of
the corresponding C*-algebras.Comment: A mistake in the previous versio
Endpoint bounds of square functions associated with Hankel multipliers
We prove endpoint bounds for the square function associated with radial
Fourier multipliers acting on radial functions. This is a consequence
of endpoint bounds for a corresponding square function for Hankel multipliers.
We obtain a sharp Marcinkiewicz-type multiplier theorem for multivariate Hankel
multipliers and bounds of maximal operators generated by Hankel
multipliers as corollaries. The proof is built on techniques developed by
Garrig\'{o}s and Seeger for characterizations of Hankel multipliers.Comment: 26 page
Schur multipliers of Cartan pairs
We define the Schur multipliers of a separable von Neumann algebra M with
Cartan masa A, generalising the classical Schur multipliers of . We
characterise these as the normal A-bimodule maps on M. If M contains a direct
summand isomorphic to the hyperfinite II_1 factor, then we show that the Schur
multipliers arising from the extended Haagerup tensor product are strictly contained in the algebra of all Schur multipliers
Quasi-multipliers of Hilbert and Banach C*-bimodules
Quasi-multipliers for a Hilbert C*-bimodule V were introduced by Brown, Mingo
and Shen 1994 as a certain subset of the Banach bidual module V**. We give
another (equivalent) definition of quasi-multipliers for Hilbert C*-bimodules
using the centralizer approach and then show that quasi-multipliers are, in
fact, universal (maximal) objects of a certain category. We also introduce
quasi-multipliers for bimodules in Kasparov's sense and even for Banach
bimodules over C*-algebras, provided these C*-algebras act non-degenerately. A
topological picture of quasi-multipliers via the quasi-strict topology is
given. Finally, we describe quasi-multipliers in two main situations: for the
standard Hilbert bimodule l_2(A) and for bimodules of sections of Hilbert
C*-bimodule bundles over locally compact spaces.Comment: 19 pages v2: to appear in Math. Scand., small glitches in one example
and with formulation of definition correcte
- âŠ