Banach algebra structure on homogeneous spaces

This paper presents a systematic theoretical study for the abstract notion of operator-valued Fourier transforms over homogeneous spaces of compact groups. Let G be a compact group, H be a closed subgroup of G , and \mu be the normalized G -invariant measure over the left coset space G/H associated to the Weil's formula. We introduce the generalized notions of abstract dual homogeneous space \widehat{G/H} for the compact homogeneous space G/H and also the operator-valued Fourier transform over the Banach function space L^1(G/H,\mu) . We prove that the abstract Fourier transform over G/H satisfies the Plancherel formula and the Poisson summation formula.

In this thesis, we establish some ergodic theorems related to Ap(G), the Figà-Talamanca-Herz algebra of a locally compact group G. This thesis is divided in two main portions. The first part is primarily concerned with the study of ergodic sequences in Ap(G) and with a newly introduced notion of ergodic multipliers. After presenting a full description of the non-degenerate *-representations of Ap(G) and of their extensions to the multiplier algebra MAp(G), it is shown that, for all locally compact groups, the weakly ergodic sequences in MAp(G) coincide with the strongly ergodic ones, and that they are, in a sense, approximating sequences for the topologically invariant means on some spaces of linear functionals on Ap(G). Next, motivated by the study of ergodic sequences of iterates, we introduce a notion of ergodic multipliers, and we provide a solution to the dual version of the complete mixing problem for probability measures, The second part is of a more abstract nature and deals with some ergodic and fixed point properties of ϕ-amenable Banach algebras. Among other things, we prove a mean ergodic theorem, establish the uniqueness of a two-sided ϕ-mean on the weakly almost periodic functionals, and provide a simpler proof of a fixed point theorem which is well known in the context of semigroups. We also study the norm spectrum of some linear functionals on Ap(G) and present a new characterization of discrete groups.

The purpose of this paper is to present some old and recent results for the class of $F$-algebras which include most classes of Banach algebras that are important in abstract harmonic analysis. We also introduce a subclass of the class of $F$-algebras, called normal $F$-algebras, that captures better the measure algebras and the (reduced) Fourier-Stieltjes algebras, and use this to give new characterisations the reduced Fourier-Stieltjes algebras of discrete groups.

We prove that the existence of a solution operator for a convolution operator from the space of ultradifferntiable functions to the corresponding space of ultradistributions is equivalent to the existence of a continuous solution operator in the space of functions. Our results are in the spirit of a classical characterization of the surjectivity of convolution operators due to Hormander. The behaviour of a fixed convolution operator in different classes of ultradifferentiable functions of Beurling type concerning the existence of a continuous linear right inverse is also considered.

Metric Projections in Spaces of Integrable Functions

9R7. Classical Many-Body Problems Amenable to Exact Treatments. (Solvable and/or Integrable and/or Linearizable…) in One-, Two-, and Three-Dimensional Space. Lecture Notes in Physics, Vol m66. - F Calogero (Dept of Phys, Univ of Rome “La Sapienza,” p Aldo Moro, Rome, 00185, Italy). Springer-Verlag, Berlin. 2001. 749 pp. ISBN 3-540-41764-8. $79.95.Reviewed by M Pascal (Lab de Modelisation en Mec, Univ Pierre et Marie Curie, Tour 66, 4 Place Jussieu, Paris, 75252 Cedex 05, France).This book is concerned with integrable problems in classical mechanics (excluding quantam or relativistic mechanics). These problems are related to the motion of a system of particles acted upon by several kinds of forces in one-, two-, or three-dimensions. More general Hamiltonian systems (not associated with a Lagrangian function) are also considered. In most cases, integrable Hamiltonian systems mean Liouville integrable systems. In almost all examples dealing with the motion of a set of particles, the masses are the same for all particles. Several methods to find integrable problems are described, but the main part of the book is devoted to the Lax pair technique. This book provides a rather exhaustive survey of results obtained in the past by scientists such as Moser, Toda, Flashka, and also by the author himself.The book involves five chapters followed by several appendices and a rather wide list of references. Each chapter includes a great amount of examples and exercises. Due to the rather abstract topics studied in the book, no figures are included in it. The first chapter is a short introduction giving some basic knowledge about Newtonian equations of motion, Hamiltonian formalism, and Liouville integrable systems. The main part of the book is contained in Chapter 2, which deals with one-dimensional motions of particles along a line or a circle. Liouville integrable problems are found by using the Lax pair technique. Several examples are obtained by solving three kinds of functional equations. From these integrable cases, other integrable systems are deduced by means of several transformations like change of variables, duplication and so on. Another method introduces a time dependant polynomial of degree N with respect to the variable X; the N zeros of this polynomial are interpreted as the positions of N particles (with the same masses) on a line. Assuming that the polynomial is solution of a linear partial derivative equation, several solvable many body problems are found. An extension of this approach is also presented, using the exact Lagrangian interpolation method. In Chapter 3, a generalized formulation of this last method is formulated for spaces of arbitrary dimensions. Chapter 4 deals with integrable systems in the plane. The method is based on the idea that integrable problems in two-dimensional space can be deduced from integrable problems in one-dimension assuming that the equations of motion hold also for complex values of the variables: it is the so called technique of complexification. A survey of integrable systems in the plane obtained by this method is given at the end of the chapter. The last chapter presents integrable many body systems in three-dimensional space. The idea is to identify some solvable evolution equations governing a time dependant matrix with the motion equations of a set of particles in three-dimensions. Again, several examples are presented. As a conclusion, Classical Many-Body Problems Amenable to Exact Treatments…, is not intended for a very large audience, but it provides a rather exhaustive survey about integrable problems in some special cases of particle dynamics. It can be useful for researchers involved in celestial mechanics or can be used as a background for teaching an undergraduate course.

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

Of concern is the unbounded operatorAΦf=f′{A_\Phi }f = f’with nonlinear domainD(AΦ)={f∈W1,1:f(0)=Φ(f)}D({A_\Phi }) = \{ f \in {W^{1,1}}:f(0) = \Phi (f)\}which is considered on the Banach spaceEEof Bochner integrable functions on an interval with values in a Banach spaceFF. Under the assumption thatΦ\Phiis a Lipschitz continuous operator fromEEtoFF, it is shown thatAΦ{A_{\Phi }}generates a strongly continuous translation semigroup(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}. For linear operatorsΦ\Phiseveral properties such as essential-compactness, positivity, and irreducibility of the semigroup(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}depending on the operatorΦ\Phiare studied. It is shown that ifFFis a Banach lattice with order continuous norm, then(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}is the modulus semigroup of(TΦ(t))t≥0{({T_\Phi }(t))_{t \geq 0}}. Finally spectral properties ofAΦ{A_\Phi }are studied and the spectral bounds(AΦ)s({A_\Phi })is determined. This leads to a result on the global asymptotic behavior in the case whereΦ\Phiis linear and to a local stability result in the case whereΦ\Phiis Fréchet differentiable.

The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications.

Let G be a compact abelian group with dual group Γ. For 1 ≤ p < ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to lp(Γ). We investigate several problems related to multipliers from Ap(G) to Aq(G). In particular, we prove that (Ap, Ap) ⊊ (Aq, Aq). For the circle group, we characterise permutation invariant multipliers from Ap to Ar for 1 ≤ r ≤ 2.

An evolution inclusion with time-dependent family of maximal monotone operators in considered in a separable Hilbert space. If the elements with minimum norm of the family of maximal monotone operators satisfy certain growth conditions, then the domains of definition of this family are closed convex sets. Hence the sweeping process is well defined, whose values are the normal cones of the domains of definition of maximal monotone operators. It is shown that if the sweeping process has a solution for each single-valued perturbation from the space of integrable functions, then the evolution inclusion with the maximal monotone operators and single-valued perturbations from the space of integrable functions is also solvable. Quite general conditions in terms of the properties of the family of maximal monotone operators that ensure the existence of solutions for the sweeping process are presented. All results obtained and the approach presented are new. They are used to prove an existence theorem for evolution inclusions with multivalued perturbations, whose values are closed nonconvex sets. Bibliography: 19 titles.

Dense Sets in Spaces of Integrable Functions

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

1. Introduction.Let S be a locally compact semigroup with a dense maximal subgroup G.We ask the following question: When is G a topological group under the induced topology?It was proved recently by P. S. Mostert [8] that in general G is not a topological group.However, there are several good reasons motivating the study of such a problem.First, for an arbitrary locally compact semigroup, the closure of the maximal subgroup containing an idempotent forms a locally compact semigroup.If we know this maximal subgroup is a topological group, then it is a locally compact topological group.(This fact will be proved in the following paragraph.)By our rather good knowledge of locally compact topological groups, we can obtain information about the structure of this locally compact semigroup.Second, if we look at this problem from the point of view of transformation groups, we have a group G acting on a locally compact topological space.Naturally, we would like this group to be topologized so that it is a topological group.It is known that the weakest topology for G to be a topological group is the g-topology [1].It is somewhat harder to deal with the g-topology than with the fc-topology, but under some conditions the ^-topology will coincide with the fe-topology.Among the known conditions, the local compactness of the group seems most interesting.If one tries to generalize such a condition, it is rather natural to ask the following : If G is a group of homeomorphisms of X onto X, and the closure of G is locally compact in the fc-topology, does the induced topology of G coincide with the g-topology?Third, in the past few years there have been several papers about such questions : What is the relationship between the algebraic structure and the topological structure?When can one strengthen the topological structure so that even under weak conditions the algebraic operations will be continuous in a certain sense?For instance, in topological groups, if the group space is locally compact, and if the multiplication is separately continuous with respect to the topology of the space, then the group is a topological group.We are able to prove the same theorem when the space is second countable and second category without the assumption of local compactness [9].Now, our present problem arises quite

The purpose of this paper is to introduce and study a function space A_(α,w)^(B,Y) (R^d ) to be a linear space of functions h∈L_w^1 (R^d ) whose fractional Fourier transforms F_α h belong to the Wiener-type space W(B,Y)(R^d ), where w is a Beurling weight function on R^d. We show that this space becomes a Banach algebra with the sum norm 〖‖h‖〗_(1,w)+〖‖F_α h‖〗_(W(B,Y)) and Θ convolution operation under some conditions. We find an approximate identity in this space and show that this space is an abstract Segal algebra with respect to L_w^1 (R^d ) under some conditions.