Let G be a locally compact group. Consider the Banach algebra L_1(G)^**, equipped with the first Arens multiplication, as well as the algebra LUC(G)^*, the dual of the space of bounded left uniformly continuous functions on G, whose product extends the convolution in the measure algebra M(G). We present (for the most interesting case of a non-compact group) completely different - in particular, direct - proofs and even obtain sharpened versions of the results, first proved by Lau-Losert and Lau, that the topological centres of the latter algebras precisely are L_1(G) and M(G), respectively. The special interest of our new approach lies in the fact that it shows a fairly general pattern of solving the topological centre problem for various kinds of Banach algebras; in particular, it avoids the use of any measure theoretical techniques. At the same time, deriving both results in perfect parallelity, our method reveals the nature of their close relation.
We refine Amitsur's theory of radicals in complete lattices and apply the obtained results to the theory of radicals in the lattices of subspaces of Banach spaces and in the lattices of ideals of Banach and C*-algebras and of Banach Lie algebras.
In this paper, we investigate the existence and uniqueness of $(ω,ρ)$-periodic solutions for a class of the abstract impulsive integro-differential equations on Banach space.
We prove results of existence of a solution (resp. existence and uniqness of a solution) for nonlinear differential equations of type $x'(t) +G(x,t) x(t) = F(x,t),$ in an abstract Banach subspace $X$ of the space of bounded real-valued continuous functions, satisfying some general and natural property. In our work, the functions $F$ and $G$ jointly depend on the variables $(x,t)\in X\times \R$. Several examples will be given, in various function spaces, to illustrate our results. The vector-valued framework is also considered.
We study multiplier algebras for a large class of Banach algebras which contains the group algebra $L_1(G)$, the Beurling algebras $L_1(G, ω)$, and the Fourier algebra $A(G)$ of a locally compact group $G$. This study yields numerous new results and unifies some existing theorems on $L_1(G)$ and $A(G)$ through an abstract Banach algebraic approach. Applications are obtained on representations of multipliers over locally compact quantum groups and on topological centre problems. In particular, five open problems in abstract harmonic analysis are solved.
We propose a unifying approach to many approximation properties studied in the literature from the 1930s up to our days. To do so, we say that a Banach space E has the (I,J,τ)-approximation property if E-valued operators belonging to the operator ideal I can be approximated, with respect to the topology τ, by operators belonging to the operator ideal J. Restricting τ to a class of linear topologies, which we call ideal topologies, this concept recovers many classical/recent approximation properties as particular instances and several important known results are particular cases of more general…
The notions of column and row operator space were extended by A. Lambert from Hilbert spaces to general Banach spaces. In this paper, we use column and row spaces over quotients of subspaces of general $L_p$-spaces to equip several Banach algebras occurring naturally in abstract harmonic analysis with canonical, yet not obvious operator space structures that turn them into completely bounded Banach algebras. We use these operator space structures to gain new insights on those algebras.
In this paper we propose a general coorbit space theory suitable to define coorbits of quasi-Banach spaces using an abstract continuous frame, indexed by a locally compact Hausdorff space, and an associated generalized voice transform. The proposed theory realizes a further step in the development of a universal abstract theory towards various function spaces and their atomic decompositions which has been initiated by Feichtinger and Gr{ö}chenig in the late 1980ies. We combine the recent approaches in Rauhut, Ullrich and Rauhut to identify, in particular, various inhomogeneous (quasi-Banach) s…
This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice $E$. We prove that the class $\mathscr D$ of regular operators for which the diagonal coincides with the atomic diagonal is always a band in $\mathcal L_r(E)$, which contains the band of abstra…
We continue the investigation of abstract Uryson operators in vector lattices. Using the recently proved Up-and-down theorem for order bounded, orthogonally additive operators, we consider the domination problem for AM-compact abstract Uryson operators. We obtain the Dodds-Fremlin type theorem and prove that for an AM- compact positive abstract Uryson operator T from a Banach lattice E to a order continuous Banach lattice F, every abstract Uryson operator S from E to F included between 0 and T is also AM-compact.
In this paper we introduce the notion of $Δ$-weak character amenable Banach algebras and investigate $Δ$-weak character amenability of certain Banach algebras such as projective tensor product $A\widehat{\otimes}B$, Lau product $A\times_TB$ for every Banach algebra $ A$ and $B$, where $T$ be a homomorphism from $B$ into $A$, abstract Segal algebras and module extension Banach algebras.
We provide a convenient framework for the study of the well-posedness of a variety of abstract (integro)differential equations in general Banach function spaces. It allows us to extend and complement the known theory on the maximal regularity of such equations. More precisely, by methods of harmonic analysis, we identify large classes of Banach spaces which are invariant with respect to distributional Fourier multipliers. Such classes include general vector-valued Banach function spaces $Φ$ and/or the scales of Besov and Triebel-Lizorkin spaces defined by $Φ$. We apply this result to the study…
In \cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the space $\mathcal M_r$ of regular martingales on a vector lattice and the space $M_r$ of bounded regular martingales on a Banach lattice. In this note, we study these two spaces from the vector lattice point of view. We show, in particular, that these spaces need not be vector lattices. However, if the underlying space is order complete then $\mathcal M_r$ is a vector lattice and $M_r$ is a Banach lattice under the regular norm.
This paper presents a systematic operator theory approach for abstract structure of Banach measure algebras over coset spaces of compact subgroups. Let $H$ be a compact subgroup of a locally compact group $G$ and $G/H$ be the left coset space associated to the subgroup $H$ in $G$. Also, let $M(G/H)$ be the Banach measure space consists of all complex measures over $G/H$. We then introduce an operator theoretic characterization for the abstract notion of involution over the Banach measure space $M(G/H)$.
Let $A$ and $B$ be Banach algebras, $θ: A\to B$ be a continuous Banach algebra homomorphism and $I$ be a closed ideal in $B$. Then the direct sum of $A$ and $I$ with respect to $θ$, denoted $A\bowtie^θI$, with a special product becomes a Banach algebra which is called the amalgamated Banach algebra. In this paper, among other things, we compute the topological centre of $A\bowtie^θI$ in terms of that of $A$ and $I$. Using this, we provide a characterization of the Arens regularity of $A\bowtie^θI$. Then we determine the character space of $A\bowtie^θI$ in terms of that of $A$ and $I$. Moreover…
For a commutative semi-simple Banach algebra ${A}$ which is an ideal in its second dual we give a necessary and sufficient condition for an essential abstract Segal algebra in ${A}$ to be a BSE-algebra. We show that a large class of abstract Segal algebras in the Fourier algebra $A(G)$ of a locally compact group $G$ are BSE-algebra if and only if they have bounded weak approximate identities. Also, in the case that $G$ is discrete we show that $A_{\rm cb}(G)$ is a BSE-algebra if and only if $G$ is weakly amenable. We study the BSE-property of some certain Segal algebras implemented by local fu…
We provide a survey of the similarities and differences between Banach and Frechet algebras including some known results and examples. We also collect some important generalizations in abstract harmonic analysis; for example, the generalization of the concepts of vector-valued Lipschitz algebras, abstract Segal algebras, Arens regularity, amenability, weak amenability, ideal amenability, etc.
An L-embedded Banach spaace is a Banach space which is complemented in its bidual such that the norm is additive between the two complementary parts. On such spaces we define a topology, called an abstract measure topology, which by known results coincides with the usual measure topology on preduals of finite von Neumann algebras (like $L_1([0,1])$). Though not numerous, the known properties of this topology suffice to generalize several results on subspaces of $L_1([0,1])$ to subspaces of arbitrary L-embedded spaces.
In this paper we provide an abstract aproach to the study of classes of multiple summing multilinear operators between Banach spaces. The main purpose is unify the study of several known classes and results, for example multiple $(p, q_1,\dots, q_n)$-summing operators, multiple mixing $(s, q, p)$-summing operators and multiple strong $(s, q, p)$-mixing summing operators. We also define new class of multiple summing multilinear operator that are particular cases of our construction and, therefore, satisfy the results proved in the paper.
We prove an abstract interpolation theorem which interpolates the (r,2)-summing and (s,2)-mixing norm of a fixed operator in the image and the range space. Combined with interpolation formulas for spaces of operators we obtain as an application the original Bennett-Carl inequalities for identities acting between Minkowski spaces l_u as well as their analogues for Schatten classes S_u. Furthermore, our techniques motivate a study of Bennett-Carl inequalities within a more general setting of symmetric Banach sequence spaces and unitary ideals.
We give an abstract definition, similar to the axioms of a Stein manifold, of a class of complex Banach manifolds in such a way that a manifold belongs to the class if and only if it is biholomorphic to a closed split complex Banach submanifold of a separable Banach space.
We extend an example of B. Aupetit, which illustrates spectral discontinuity for operators on an infinite dimensional separable Hilbert space, to a general spectral discontinuity result in abstract Banach algebras. This can then be used to show that given any Banach algebra, $Y$, one may adjoin to $Y$ a non-commutative inessential ideal, $I$, so that in the resulting algebra, $A$, the following holds: To each $x\in Y$ whose spectrum separates the plane there corresponds a perturbation of $x$, of the form $z=x+a$ where $a\in I$, such that the spectrum function on $A$ is discontinuous at $z$.
We extend some results about sampling of entire functions of exponential type to Banach spaces. By using generator $D$ of one-parameter group $e^{tD}$ of isometries of a Banach space $E$ we introduce Bernstein subspaces $\mathbf{B}_σ(D),\>\>σ>0,$ of vectors $f$ in $E$ for which trajectories $e^{tD}f$ are abstract-valued functions of exponential type which are bounded on the real line. This property allows to reduce sampling problems for $e^{tD}f$ with $f\in \mathbf{B}_σ(D)$ to known sampling results for regular functions of exponential type $σ$.
There are two main aims of the paper. The first one is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second one is to extend the criterion for the precompactness of sets in the Lebesgue spaces $L_p(\mathbb R^n)$, $1 \leq p < \infty$, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces $B^s_{p,q}(\mathbb R^n)$, $0<s<1$, $0<p,q\leq \in…
Let $\{F_n\}$ be the sequence of the Fejér kernels on the unit circle $\mathbb{T}$. The first author recently proved that if $X$ is a separable Banach function space on $\mathbb{T}$ such that the Hardy-Littlewood maximal operator $M$ is bounded on its associate space $X'$, then $\|f*F_n-f\|_X\to 0$ for every $f\in X$ as $n\to\infty$. This implies that the set of analytic polynomials $\mathcal{P}_A$ is dense in the abstract Hardy space $H[X]$ built upon a separable Banach function space $X$ such that $M$ is bounded on $X'$. In this note we show that there exists a separable weighted $L^1$ space…
Given a fully symmetric Banach function space $E$ and a decreasing positive weight $w$ on $I = (0, a)$, $0 < a \le \infty $, the generalized Lorentz space $Λ_{E,w}$ is defined as the symmetrization of the canonical copy $E_w$ of $E$ on the measure space associated with the weight. If $E$ is an Orlicz space then $Λ_{E,w}$ is an Orlicz-Lorentz space. An investigation of the Köthe duality of these classes is developed that is parallel to preceding works on Orlicz-Lorentz spaces. First a class of functions $M_{E,w}$, which does not need to be even a linear space, is similarly defined as the symme…
We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of $\mc A$ is Arens regular, and give some evidence that this is if and only if $\mc A$ is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebra. We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of $\mc A$ is amenable. We provide an abstract cha…
In this paper we give sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces. The main technique is the use of algebraic structure (for usual convolution product $\ast$) of these solutions which are defined by a version of the Duhamel formula. We define algebra homomorphisms from a new class of test-functions and apply our results to concrete operators. Finally, we introduce the notion of $k$-distribution semigroups to extend previous concepts of distribution semigroups.
In this paper, we construct the abstract ideal of polynomials. We show this is an ideal of Banach and, in a second moment, we explore the question of the coherence and compatibility of the pair composed by the abstract ideals of polynomials and multilinear applications.
In this paper, we deal with the existence of $ω$-periodic mild solutions for the abstract evolution equation with delay in an ordered Banach space $E$ $$u'(t)+Au(t)=F(t,u(t),u(t-τ)),\ \ \ \ t\in\R,$$ where $A:D(A)\subset E\rightarrow E$ is a closed linear operator and $-A$ generates a positive $C_{0}$-semigroup $T(t)(t\geq0)$, $F:\R\times E\times E\rightarrow E$ is a continuous mapping which is $ω$-periodic in $t$, and $τ\geq0$ is a constant. Under some weaker assumptions, we construct monotone iterative method for the delayed evolution equation periodic problems, and obtain the existence of m…
We extend Pisier's abstract version of Grothendieck's theorem to general non-locally convex quasi-Banach spaces. We also prove a related result on factoring operators through a Banach space and apply our results to the study of vector-valued inequalities for Sidon sets. We also develop the local theory of (non-locally convex) spaces with duals of weak cotype 2.
We present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr and Bochner type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework.
The Banach algebra LUC(G)* associated to a topological group G has been of interest in abstract harmonic analysis. A number of authors have studied the topological centre of LUC(G)*, which is defined as the set of elements in LUC(G)* for which the left multiplication is w*--w*-continuous on LUC(G)*. Several recent works show that for a locally compact group G it is sufficient to test the continuity of the left multiplication at just one specific point in order to determine whether an element of LUC(G)* belongs to the topological centre. In this work we extend some of these results to a much la…
In this paper, first-order Sobolev-type spaces on abstract metric measure spaces are defined using the notion of (weak) upper gradients, where the summability of a function and its upper gradient is measured by the "norm" of a quasi-Banach function lattice. This approach gives rise to so-called Newtonian spaces. Tools such as moduli of curve families and Sobolev capacity are developed, which allows us to study basic properties of these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are established.
Let ${\mathcal A}$ and ${\frak A}$ be Banach algebras such that ${\mathcal A}$ is a Banach ${\frak A}$-bimodule with compatible actions. We define the product ${\cal A}\rtimes{\frak A}$, which is a strongly splitting Banach algebra extension of ${\frak A}$ by $\cal A$. After characterization of the multiplier algebra, topological centre, (maximal) ideals and spectrum of ${\cal A}\rtimes{\frak A}$, we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of ${\cal A}\rtimes{\frak A}$ in relation to that of the algebras $\cal A$, $\frak A$ and the action of $\fr…
Associated with a locally compact group $\cal G$ and a $\cal G$-space $\cal X$ there is a Banach subspace $LUC({\cal X},{\cal G})$ of $C_b({\cal X})$, which has been introduced and studied by Lau and Chu in \cite{chulau}. In this paper, we study some properties of the first dual space of $LUC({\cal X},{\cal G})$. In particular, we introduce a left action of $LUC({\cal G})^*$ on $LUC({\cal X},{\cal G})^*$ to make it a Banach left module and then we investigate the Banach subalgebra ${\frak{Z}({\cal X},{\cal G})}$ of $LUC({\cal G})^*$, as the topological centre related to this module action, whi…
We analyze a definition of product of Banach spaces that is naturally associated by duality with an abstract notion of space of multiplication operators. This dual relation allows to understand several constructions coming from different fields of the functional analysis, that can be seen as instances of the abstract one when a particular product is considered. Some relevant examples and applications are shown.
We study the asymptotic behavior of almost-orbits of abstract evolution systems in Banach spaces with or without a Lipschitz assumption. In particular, we establish convergence, convergence in average and almost-convergence of almost-orbits both for the weak and the strong topologies based on the behavior of the orbits. We also analyze the set of almost-stationary points.
Given a Banach space operator $A$, the isolated eigenvalues $E(A)$ and the poles $Π(A)$ (resp., eigenvalues $E^a(A)$ which are isolated points of the approximate point spectrum and the left ploles $Π^a(A)$) of the spectrum of $A$ satisfy $Π(A)\subseteq E(A)$ (resp., $Π^a(A)\subseteq E^a(A)$), and the reverse inclusion holds if and only if $E(A)$ (resp., $E^a(A)$) has empty intersection with the B-Weyl spectrum (resp., upper B-Weyl spectrum) of $A$. Evidently $Π(A)\subseteq E^a(A)$, but no such inclusion exists for $E(A)$ and $Π^a(A)$. The study of identities $E(A)=Π^a(A)$ and $E^a(A)=Π(A)$, an…
In this paper we present an abstract framework for construction of Banach spaces of distributions from group representations. This generalizes the theory of coorbit spaces initiated by H.G. Feichtinger and K. Gröchenig in the 1980's. Spaces that can be described by this new technique include the whole Banach-scale of Bergman spaces on the unit disc. For these Bergman spaces we show that atomic decompositions can be constructed through sampling. We further present a wavelet characterization of Besov spaces on the forward light cone.
In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted matrices boundedness below of the corresponding operator in $\ell^p$ for some $p$ implies boundedness below in $\ell^p$ for all $p$. We use the established resultto enrich our understanding of Banach frames and obtain new results for irregular sampling problems. We also present a version of a non-commutative Wiener's lemma for slanted matrices.
We refer to the real Jordan Banach algebra of bounded Hermitian operators on a Hilbert space as a Hermitian algebra. We define an abstract Hermitian algebra (AH-algebra) to be the directed group of an e-ring that contains a semitransparent element, has the quadratic annihilation property, and satisfies a Vigier condition on pairwise commuting ascending sequences. All of this terminology is explicated in this article, where we launch a study of AH-algebras. Here we establish the fundamental properties of AH-algebras, including the existence of polar decompositions and spectral resolutions, and …
Boundary value problem for complete second order elliptic equation is considered in Banach space. The equation and boundary conditions involve a small and spectral parameter. The uniform L_{p}-regularity properties with respect to space variable and parameters are established. Here, the explicit formula for the solution is given and behavior of solution is derived when the small parameter approaches zero. It used to obtain singular perturbation result for abstract elliptic equation
We establish a general CCR (liminarity) property for uniformly bounded irreducible representations of nilpotent Lie groups on reflexive Banach spaces, extending the well known property of unitary irreducible representations of these groups on Hilbert spaces. We also prove that this conclusion fails for many representations on non-reflexive Banach spaces. Our approach to these results blends the method of transference from abstract harmonic analysis and a systematic use of spaces of smooth vectors with respect to Lie group representations.
We study the cohomology of isometric group actions on (super) reflexive Banach spaces with a focus on the relation between finite conjugacy classes and split exactness of cochain complexes. In particular, we show that, if a uniformly convex Banach module has no almost invariant vectors under the FC-centre of the acting group, then the associated cochain complex is split exact. Other similar rigidity results are established that are related to prior work of Bader - Furman - Gelander - Monod, Bader - Rosendal - Sauer and Nowak.
A variant of the abstract Cauchy-Kovalevskaya theorem is considered. We prove existence and uniqueness of classical solutions to the nonlinear, non-autonomous initial value problem \[ \frac{du(t)}{dt} = A(t)u(t) + B(u(t),t), \ \ u(0) = x \] in a scale of Banach spaces. Here $A(t)$ is the generator of an evolution system acting in a scale of Banach spaces and $B(u,t)$ obeys an Ovcyannikov-type bound. Continuous dependence of the solution with respect to $A(t)$, $B(u,t)$ and $x$ is proved. The results are applied to the Kimura-Maruyama equation for the mutation-selection balance model. This yiel…
In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition of an abstract representation ring, and representation ideal. The completion is then a commutative Banach a…
We show that a representation of a Banach algebra $A$ on a Banach space $X$ can be extended to a canonical representation of $A^{**}$ on $X$ if and only if certain orbit maps $A\to X$ are weakly compact. When this is the case, we show that the essential space of the representation is complemented if $A$ has a bounded left approximate identity. This provides a tool to disregard the difference between degenerate and nondegenerate representations. Our results have interesting consequences both in C*-algebras and in abstract harmonic analysis. For example, a C*-algebra $A$ has an isometric represe…
