Completely bounded disjointness preserving operators between Fourier algebras

In this paper, we characterize surjective completely bounded disjointness preserving linear operators on Fourier algebras of locally compact amenable groups. We show that such operators are given by weighted homomorphisms induced by piecewise affine proper maps.     

Spectral analysis for convolution operators on locally compact groups

We consider operators H_μ of convolution with measures μ on locally compact groups. We characterize the spectrum of H_μ by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of H_μ, respectively. The proofs rely on commutator methods.     

Topological centers of certain dual algebras

Let be a Banach algebra with a bounded approximate identity. Let and be, respectively, the topological centers of the algebras and . In this paper, for weakly sequentially complete Banach algebras, in particular for the group and Fourier algebras and , we study the sets , , the relations between them and with several other subspaces of or .

     

The amenability constant of the Fourier algebra

For a locally compact group , let denote its Fourier algebra and its dual object, i.e., the collection of equivalence classes of unitary representations of . We show that the amenability constant of is less than or equal to and that it is equal to one if and only if is abelian.

     

Multiplier operators and extremal functions related to the dual dunkl-sonine operator

We study the dual Dunkl-Sonine operator ^tS_k,? on ?^d, and give expression of ^tS_k,?, using Dunkl multiplier operators on ?^d. Next, we study the extremal functions f^*_λ, λ >0 related to the Dunkl multiplier operators, and more precisely show that {f^*_λ} λ >0 converges uniformly to ^tS_k,?(f) as λ → 0+. Certain examples based on Dunkl-heat and Dunkl-Poisson kernels are provided to illustrate the results.     

An approach to the spectrum structure of Dirac operators by the local-compactness method

The purpose of this paper is to investigate the spectra of the Dirac operator . The local compactness of is shown under some assumption on . This method enables us to prove that if as , then and to give a significant sufficient condition that or has a purely discrete spectrum.

     

多重调和Bergman空间上的Toeplitz算子和Hankel算子

完全刻画多重调和Bergman空间上Toeplitz算子和Hankel算子的紧性.运用紧Toeplitz算子这个结果,建立了Toeplitz代数和小Hankel代数的短正合列,推广了单位圆盘上相应的结果.     

Decomposition of

For any locally compact group , let and be the Fourier and the Fourier-Stieltjes algebras of , respectively. is decomposed as a direct sum of and , where is a subspace of consisting of all elements that satisfy the property: for any and any compact subset , there is an with and such that is characterized by the following: an element is in if and only if, for any there is a compact subset such that for all with and . Note that we do not assume the amenability of . Consequently, we have for all if is noncompact. We will apply this characterization of to investigate the general properties of and we will see that is not a subalgebra of even for abelian locally compact groups. If is an amenable locally compact group, then is the subspace of consisting of all elements with the property that for any compact subset , .

     

Representations, convolution square roots and spherical vectors

We extend results on square integrable representations of a locally compact group to subrepresentations of a representations induced from a character of a closed subgroup. We do so working in the framework of quotient representations of ∗-algebras.     

Power boundedness in Fourier and Fourier-Stieltjes algebras and other commutative Banach algebras

We study power boundedness in the Fourier and Fourier-Stieltjes algebras, A(G) and B(G), of a locally compact group G as well as in some other commutative Banach algebras. The main results concern the question of when all elements with spectral radius at most one in any of these algebras are power bounded, the characterization of power bounded elements in A(G) and B(G) and also the structure of the Gelfand transform of a single power bounded element.     

Uniform primeness of the Jordan algebra of symmetric operators

In this note we establish the best possible constant for the general lower estimate for the Jacobson - McCrimmon operator on the algebra of symmetric operators acting on a Hilbert space.

     

Characterizations of elements with compact support in the dual spaces of -modules of

For a locally compact group and , let be the Figà-Talamanca-Herz algebra and let be its dual Banach space. For a Banach -module of , we denote the norm closure of the subspace of the elements in with compact support by . We prove that an element of is in if and only if for any 0$">, there exists a compact subset of such that for all with and . In particular, we have that an element of is in if and only if for any 0$">, there exists a compact subset of such that for all with . If has an orthogonal complement in , we characterize by the following condition: is in if and only if for any 0$"> and any compact subset of , there exists some with and such that \Vert u\Vert - \epsilon $">. Some results of Flory (1971) and Miao (1999) can be obtained from our main theorems by taking and as some -subalgebras of .

     

Uniformly continuous superposition operators in the space of bounded variation functions

Let I, J ? ? be intervals. The main result says that if a superposition operator H generated by a function of two variables h: I × J → ?, H (φ)(x) ? h (x, φ (x)), maps the set BV (I, J) of all bounded variation functions, φ: I → J into the Banach space BV (I, ?) and is uniformly continuous with respect to the BV ‐norm, then h (x, y) = a (x)y + b (x), x ∈ I, y ∈ J, for some a, b ∈ BV (I, ?) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Best bounds for approximate identities in ideals of the Fourier algebra vanishing on subgroups

In this paper we show that if is an amenable locally compact group and if is a closed subgroup, then the ideal has an approximate identity of norm If is not open, this bound is the best possible.

     

Contractive projections on Banach algebras

In this paper, we explore the properties of projections of norm one on general Banach algebras, in particular the relation with conditional expectations for algebras which arise in harmonic analysis.     

On the Positive Definiteness of Certain Functions

For a coinmutative senugoup (S, +, *) with involution and a function f : S → [0, ∞), the set S(f) of those p ≥ 0 such that f^P is a positive definite function on S is a closed subsemigroup of [0, ∞) containing 0. For S = (IR, +, x* = -x) it may happen that S(f) = { kd : k ∈ N₀ } for some d > 0, and it may happen that S(f) = {0} ? [d, ∞) for some d > O. If α > 2 and if S = (?, +, n* = -n) and f(n) = e^?[n]α or S = (IN₀, +, n* = n) and f(n) = e^nα, then S(f) ∪ (0, c) = ? and [d, ∞) ? S(f) for some d ≥; c > 0. Although (with c maximal and d minimal) we have not been able to show c = d in all cases, this equality does hold if S = ? and α ≥ 3.4. In the last section we give sinipler proofs of previously known results concerning the positive definiteness of x → e^?||x||α on normed spaces.     

Extreme points of the unit ball of the Fourier-Stieltjes algebra

Let be a locally compact group. Among other things, we proved in this paper that for an IN-group , the extreme points of the unit ball of the Fourier-Stieltjes algebra are not in the Fourier algebra if and only if is non-compact, or equivalently, there is no irreducible representation of which is quasi-equivalent to a subrepresentation of the left regular representation of if and only if is non-compact. This result is a non-commutative version of the following well known result: For any locally compact group , the extreme points of the unit ball of the measure algebra are not in the group algebra if and only if is non-discrete. On the other hand, we also showed that if has the RNP, then there are extreme points of the unit ball of that are in . Since it is well known there are non-compact locally compact group for which has the RNP, there exist non-compact locally compact groups where extreme points of the unit ball of can be in . This shows that the condition be an IN-group cannot be entirely removed.