164,277 research outputs found

    Local Operator Multipliers and Positivity

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    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

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    Let GG be an infinite locally compact abelian group. If XX is Banach space, we show that if every bounded Fourier multiplier TT on L2(G)L^2(G) has the property that T\ot Id_X is bounded on L2(G,X)L^2(G,X) then the Banach space XX is isomorphic to a Hilbert space. Moreover, if 1<p<∞1<p<\infty, p=Ìž2p\not=2, we prove that there exists a bounded Fourier multiplier on Lp(G)L^p(G) 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

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    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

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    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

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    We prove endpoint bounds for the square function associated with radial Fourier multipliers acting on LpL^{p} 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 LpL^p 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

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    We define the Schur multipliers of a separable von Neumann algebra M with Cartan masa A, generalising the classical Schur multipliers of B(ℓ2)B(\ell^2). 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 A⊗ehAA \otimes_{eh} A are strictly contained in the algebra of all Schur multipliers

    Quasi-multipliers of Hilbert and Banach C*-bimodules

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    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
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