Multipliers between two operator spaces

Multipliers for p-Bessel sequences in Banach spaces

Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the c…

The Douglas property for multiplier algebras of operators

For a collection of reproducing kernels k which includes those for the Hardy space of the polydisk and ball and for the Bergman space, k is a complete Pick kernel if and only if the multiplier algebra of the Hilbert space H^2(k) associated to k has the Douglas property. Consequences for solving the operator equation AX=Y are examined.

Unbeschränkte Multiplikatoren auf Operatorräumen

We introduce unbounded multipliers on operator spaces. These multipliers generalize both, regular operators on Hilbert C*-modules and (bounded) multipliers on operator spaces. ----- Wir definieren den Begriff eines unbeschränkten Multiplikators auf Operatorräumen. Diese Multiplikatoren verallgemeinern sowohl reguläre Operatoren auf Hilbert-C*-Moduln als auch (beschränkte) Multiplikatoren auf Operatorräumen.

One-sided M-Ideals and Multipliers in Operator Spaces, I

The theory of M-ideals and multiplier mappings of Banach spaces naturally generalizes to left (or right) M-ideals and multiplier mappings of operator spaces. These subspaces and mappings are intrinsically characterized in terms of the matrix norms. In turn this is used to prove that the algebra of left adjointable mappings of a dual operator space X is a von Neumann algebra. If in addition X is an operator A--B-bimodule for $C^{*}$-algebras A and B, then the module operations on X are automatically weak$^{*}$ continuous. One sided L-projections are introduced, and analogues of various results …

框架乘子与Toeplitz算子的提升性质

The Lifting Property for Frame Multipliers and Toeplitz Operators

本文证明正符号框架乘子是coorbit空间间的Banach同构，推广提升定理至Gabor框架和Fock空间，得到调制空间与加权Fock空间的新同构结果。

Multipliers and dual operator algebras

In a previous paper we showed how the main theorems characterizing operator algebras and operator modules, fit neatly into the framework of the `noncommutative Shilov boundary', and more particularly via the left multiplier operator algebra of an operator space. As well as giving new characterization theorems, the approach of that paper allowed many of the hypotheses of the earlier theorems to be eliminated. Recent progress of the author with Effros and Zarikian now enables weak*-versions of these characterization theorems. For example, we prove a result analogous to Sakai's famous characteriz…

Basic definition and properties of Bessel multipliers

This paper introduces the concept of Bessel multipliers. These operators are defined by a fixed multiplication pattern, which is inserted between the Analysis and synthesis operators. The proposed concept unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames. The basic properties of this class of operators are investigated. In particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified. As a s…

Multipliers of the Hardy space H^1 and power bounded operators

We study the space of functions $ϕ\colon \NN\to \CC$ such that there is a Hilbert space $H$, a power bounded operator $T$ in $B(H)$ and vectors $ξ,η$ in $H$ such that $$ϕ(n) = < T^nξ,η>.$$ This implies that the matrix $(ϕ(i+j))_{i,j\ge 0}$ is a Schur multiplier of $B(\ell_2)$ or equivalently is in the space $(\ell_1 \buildrel {\vee}\over {\otimes} \ell_1)^*$. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of $H^1$ which we call ``shift-bounded''. We show that there is a $ϕ$ which is a ``compl…

Quasi-multipliers and algebrizations of an operator space

Let $X$ be an operator space, let $ϕ$ be a product on $X$, and let $(X,ϕ)$ denote the algebra that one obtains. We give necessary and sufficient conditions on the bilinear mapping $ϕ$ for the algebra $(X,ϕ)$ to have a completely isometric representation as an algebra of operators on some Hilbert space. In particular, we give an elegant geometrical characterization of such products by using the Haagerup tensor product. Our result makes no assumptions about identities or approximate identities. Our proof is independent of the earlier result of Blecher-Ruan-Sinclair that solved the case when the …

Operator amenability of the Fourier algebra in the cb-multiplier norm

Let $G$ be a locally compact group, and let $A_\cb(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that $\cstar(G)$ is residually finite-dimensional, we show that $A_\cb(G)$ is operator amenable. In particular, $A_\cb(F_2)$ is operator amenable even though $F_2$, the free group in two generators, is not an amenable group. Moreover, we show that, if $G$ is a discrete group such that $A_\cb(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in…

On multipliers of Hilbert modules over locally C*-algebras

In this paper, we investigate the structure of the multiplier module of a Hilbert module over a locally C*-algebra and the relationship between the set of all adjointable operators from a Hilbert A-module E to a Hilbert A-module F and the set of all adjointable operators from the multiplier module M(E)of E to the multiplier module M(F) of F.

Multipliers and operator space structure of weak product spaces

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator…

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 mult…

Central and convolution Herz-Schur multipliers

We obtain descriptions of central operator-valued Schur and Herz-Schur multipliers, akin to a classical characterisation due to Grothendieck, that reveals a close link between central (linear) multipliers and bilinear multipliers into the trace class. Restricting to dynamical systems where a locally compact group acts on itself by translation, we identify their convolution multipliers as the right completely bounded multipliers, in the sense of Junge-Neufang-Ruan, of a canonical quantum group associated with the underlying group. We provide characterisations of contractive idempotent operator-…

Noncommutative Figa-Talamanca-Herz algebras for Schur multipliers

We introduce a noncommutative analogue of the Figá-Talamanca-Herz algebra $A_p(G)$ on the natural predual of the operator space $\frak{M}_{p,cb}$ of completely bounded Schur multipliers on Schatten space $S_p$. We determine the isometric Schur multipliers and prove that the space $\frak{M}_{p}$ of bounded Schur multipliers on Schatten space $S_p$ is the closure in the weak operator topology of the span of isometric multipliers.

Multipliers of operator spaces, and the injective envelope

We study the injective envelope I(X) of an operator space X, showing amongst other things that it is a self-dual C$^*-$module. We describe the diagonal corners of the injective envelope of the canonical operator system associated with X. We prove that if X is an operator $A-B$-bimodule, then A and B can be represented completely contractively as subalgebras of these corners. Thus, the operator algebras that can act on X are determined by these corners of I(X) and consequently bimodules actions on X extend naturally to actions on I(X). These results give another characterization of the multipli…

Unconditionality, Fourier multipliers and Schur multipliers

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

Multipliers and Hereditary Subalgebras of Operator Algebras

We generalize some technical results of Glicksberg to the realm of general operator algebras and use them to give a characterization of open and closed projections in terms of certain multiplier algebras. This generalizes a theorem of J. Wells characterizing an important class of ideals in uniform algebras. The difficult implication in our main theorem is that if a projection is open in an operator algebra, then the multiplier algebra of the associated hereditary subalgebra arises as the closure of the subalgebra with respect to the strict topology of the multiplier algebra of a naturally asso…

Operator system structures and extensions of Schur multipliers

For a given C*-algebra $\mathcal{A}$, we establish the existence of maximal and minimal operator $\mathcal{A}$-system structures on an AOU $\mathcal{A}$-space. In the case $\mathcal{A}$ is a W*-algebra, we provide an abstract characterisation of dual operator $\mathcal{A}$-systems, and study the maximal and minimal dual operator $\mathcal{A}$-system structures on a dual AOU $\mathcal{A}$-space. We introduce operator-valued Schur multipliers, and provide a Grothendieck-type characterisation. We study the positive extension problem for a partially defined operator-valued Schur multiplier $φ$ and…

Projections, multipliers and decomposable maps on noncommutative $\mathrm{L}^p$-spaces

We introduce a noncommutative analogue of the absolute value of a regular operator acting on a noncommutative $\mathrm{L}^p$-space. We equally prove that two classical operator norms, the regular norm and the decomposable norm are identical. We also describe precisely the regular norm of several classes of regular multipliers. This includes Schur multipliers and Fourier multipliers on some unimodular locally compact groups which can be approximated by discrete groups in various senses. A main ingredient is to show the existence of a bounded projection from the space of completely bounded $\mat…

舒尔乘子在调和分析与算子代数中的影响

The impact of Schur multipliers in harmonic analysis and operator algebras

本文综述舒尔乘子在过去15年的影响，关注其与傅里叶乘子的联系及在调和分析和算子代数间的桥梁作用，介绍舒尔滕p类最新界在相关领域的应用，以及非托普利茨舒尔乘子等方面的新进展。

Radial multipliers on reduced free products of operator algebras

Let A_i be a family of unital C*-algebras, respectively, of von Neumann algebras and phi: N_0 \to C. We show that if a Hankel matrix related to phi is trace-class, then there exists a unique completely bounded map M_phi on the reduced free product of the A_i, which acts as an radial multiplier. Hereby we generalize a result of Wysoczański for Herz-Schur multipliers on reduced group C*-algebras for free products of groups.

Algebras of differential operators on Lie groups and spectral multipliers

This thesis is devoted to the study of joint spectral multipliers for a system of pairwise commuting, self-adjoint left-invariant differential operators L_1,...,L_n on a connected Lie group G. Under the assumption that the algebra generated by L_1,...,L_n contains a weighted subcoercive operator - a notion due to ter Elst and Robinson (J. Funct. Anal., 157(1):88--163, 1998), including positive elliptic operators, sublaplacians and Rockland operators - we prove that L_1,...,L_n are (essentially) self-adjoint and strongly commuting on L^2(G). Moreover, we perform an abstract study of such a syst…

The classification problem for operator algebraic varieties and their multiplier algebras

We study from the perspective of Borel complexity theory the classification problem for multiplier algebras associated with operator algebraic varieties. These algebras are precisely the multiplier algebras of irreducible complete Nevanlinna-Pick spaces. We prove that these algebras are not classifiable up to algebraic isomorphism using countable structures as invariants. In order to prove such a result, we develop the theory of turbulence for Polish groupoids, which generalizes Hjorth's turbulence theory for Polish group actions. We also prove that the classification problem for multiplier al…

On the C*-algebra Generated by Toeplitz Operators and Fourier Multipliers on the Hardy Space of a Locally Compact Group

Let $G$ be a locally compact abelian Hausdorff topological group which is non-compact and whose Pontryagin dual $Γ$ is partially ordered. Let $Γ^{+}\subsetΓ$ be the semigroup of positive elements in $Γ$. The Hardy space $H^{2}(G)$ is the closed subspace of $L^{2}(G)$ consisting of functions whose Fourier transforms are supported on $Γ^{+}$. In this paper we consider the C*-algebra $C^{*}(\mathcal{T}(G)\cup F(C(\dot{Γ^{+}})))$ generated by Toeplitz operators with continuous symbols on $G$ which vanish at infinity and Fourier multipliers with symbols which are continuous on one point compactific…

Which multiplier algebras are $W^*$-algebras?

We consider the question of when the multiplier algebra $M(\mathcal{A})$ of a $C^*$-algebra $\mathcal{A}$ is a $ W^*$-algebra, and show that it holds for a stable $C^*$-algebra exactly when it is a $C^*$-algebra of compact operators. This implies that if for every Hilbert $C^*$-module $E$ over a $C^*$-algebra $\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $ W^*$-algebra, then $\mathcal{A}$ is a $C^*$-algebra of compact operators. Also we show that a unital $C^*$-algebra $\mathcal{A}$ which is Morita equivalent to a $ W^*$-algebra must be a $ W^*$-algebra.

Noncommutative complex analysis and Bargmann-Segal multipliers

We state several equivalent noncommutative versions of the Cauchy-Riemann equations and characterize the unbounded operators on L^2(R) which satisfy them. These operators arise from the creation operator via a functional calculus involving a class of entire functions, identified by Newman and Shapiro [D. J. Newman and H. S. Shapiro, Fischer spaces of entire functions, in Entire Functions and Related Parts of Analysis (J. Koorevaar, ed.), AMS Proc. Symp. Pure Math. XI (1968), 360-369], which act as unbounded multiplication operators on Bargmann-Segal space.