Embedding a topological group into its WAP-compactification

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let G \mathcal {G} be a topological group, and A \mathcal {A} a unital symmetric C ∗ C^* -subalgebra of U C ( G ) \mathrm {UC}(\mathcal {G}) , the algebra of bounded uniformly continuous functions on G \mathcal {G} . Generalizing the notion of a stable metric, we study A \mathcal {A} -metrics δ \delta , i.e., the function δ ( e , ⋅ ) \delta (e, \cdot ) belongs to A \mathcal {A} ; the case A = W A P ( G ) \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}) , the algebra of weakly almost periodic functions on G \mathcal {G} , recovers stability. If the topology of G G is induced by a left invariant metric d d , we prove that A \mathcal {A} determines the topology of G \mathcal {G} if and only if d d is uniformly equivalent to a left invariant A \mathcal {A} -metric. As an application, we show that the additive group of C [ 0 , 1 ] C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [ Topological transformation groups: selected topics , Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now G \mathcal {G} be a metric group, and assume A ⊆ L U C ( G ) \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}) , the algebra of bounded left uniformly continuous functions on G \mathcal {G} , is a unital C ∗ C^* -algebra which is the uniform closure of coefficients of representations of G \mathcal {G} on members of F \mathscr {F} , where F \mathscr {F} is a class of Banach spaces closed under ℓ 2 \ell _2 -direct sums. We prove that A \mathcal {A} determines the topology of G \mathcal {G} if and only if G \mathcal {G} embeds into the isometry group of a member of F \mathscr {F} , equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability.

In this paper, we review indispensable properties and characterizations of almost periodic functions and asymptotically almost periodic functions in Banach spaces. Special accent is put on the Stepanov generalizations of almost periodic functions and asymptotically almost periodic functions. We also recollect some basic results regarding equi-Weyl-almost periodic functions and Weyl-almost periodic functions. The class of asymptotically Weyl-almost periodic functions, introduced in this work, seems to be not considered elsewhere even in the scalar-valued case. We actually introduce eight new classes of asymptotically almost periodic functions and analyze relations between them. In order to make a picture as complete and clear as possible, several illustrating examples and counter-examples are given. It is worth noting that the topics dealt with in this paper seem to be of an intrinsic connection with the problem of existence and uniqueness of solutions of differential and difference equations, in both determinist and stochastic cases.

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

Let B be a bounded linear operator having domain and range in a Banach space. If the second-order differential operator d2/dt2–B has a Bohr-Neugebauer type property for Bochner almost periodic functions, then any Stepanov-bounded solution of the differential equation (d2/dt2)u(t) – Bu(t) = g(t) is Bochner almost periodic, with g(t) being a Stepanov-almost periodic continuous function.

1. In view of the recent work on topological groups it is natural to consider the situation which arises when such groups act as transformation groups on various types of spaces. Such a study is begun here from the point of view of almost periodic transformation groups, the definition of which is suggested by von Neumann's paper on almost periodic functions in a group.t Compact topological transformation groups are a special case of almost periodic transformation groups, at least for a rather wide class of spaces. The paper concerns itself chiefly with the nature of the minimal closed invariant sets of such groups. There are some results for general spaces but the main results are for Euclidean spaces and more particularly for threedimensional Euclidean spaces. One of the most interesting theorems states that if a compact one-dimensional group acts on three-space in such a way that its orbits are uniformly bounded in diameter, then every point of the space is fixed under the group, so that if such a group is to act in a nontrivial manner the diameters of its orbits must be unbounded. Under some restrictions a similar theorem is proved for one-parameter almost periodic groups. Furthermore it is shown that for this latter class of groups, many of the orbits must actually be simple closed curves if they have one-dimensional closures. 2. The group considered here will be denoted by G. It will be subjected to various conditions as the occasion demands but it will always be Abelian. In case it is the group of real numbers, it will be spoken of as a one-parameter group; in case it is the real numbers reduced modulo one, it will be spoken of as the circle group. The space on which the group acts will be denoted by R. It will be specialized in various ways, but in any case it will always be a locally compact metric space. If x and y are two points of R, the distance between them will be denoted by d(x, y).

We prove that the additive group (E*, τ k (E)) of an $${\mathcal {L}_\infty}$$ -Banach space E, with the topology τ k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ k (E)) is uniformly homeomorphic to a subset of $${\ell_2^\kappa}$$ for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.

The purpose of this work is to study remotely almost periodic functions defined on the semi-axis ℝ + := [ 0 , + ∞ ) {\mathbb{R}_{+}:=[0,+\infty)} (respectively, on the real axis ℝ := ( - ∞ , + ∞ ) {\mathbb{R}:=(-\infty,+\infty)} ) with values in the Banach space 𝔅 {\mathfrak{B}} . We establish a relation between remotely almost periodic movements in the shift dynamical system ( C ( 𝕋 , 𝔅 ) , 𝕋 {(C(\mathbb{T},\mathfrak{B}),\mathbb{T}} , σ ) {\sigma)} ( 𝕋 = ℝ + {\mathbb{T}=\mathbb{R}_{+}} or ℝ {\mathbb{R}} ) in the space of continuous functions C ⁢ ( 𝕋 , 𝔅 ) C(\mathbb{T},\mathfrak{B}) and remotely almost periodic functions. Using this connection, we prove some important analytical and topological properties of remotely almost periodic functions.

Existence and algorithms for the bilevel new generalized mixed equilibrium problems in Banach spaces

Dynamics of a class of ODEs more general than almost periodic

Let X be a Banach space and suppose YX is a Banach space compactly embedded into X, and (ak) is a weakly null sequence of functionals in X � . Then there exists a sequence fng & 0 such that jan(y)j � nkykY for every n 2 N and every y 2 Y. We prove this result and we use it for the study of fast decay of Fourier coefficients in L p (T) and frame coefficients in the Hilbert setting. 1. Motivation One of the classical problems in Harmonic Analysis is the study of the relationship that exists between decay properties of the Fourier coefficientscn(f) = 1 2� R 2� 0 f(t)e int dt of a 2�-periodic function f : T ! C and its membership to several function spaces. Just to mention a few well known examples, we show the following list of results: � Riemann-Lebesgue Lemma sates that for f 2 L 1 (T) the Fourier coefficients satisfy limn!±1 jcn(f)j= 0. � Parseval's identity states that f 2 L 2 (T) if and only if fcn(f)g 2 ` 2 (Z). � For 1 2 and f 2 L p (T), then f 2 L 2 (T) and fcn(f)g 2 ` 2 (Z).

We establish a composition theorem of Stepanov almost periodic functions, and, with its help, a composition theorem of Stepanov-like pseudo almost periodic functions is obtained. In addition, we apply our composition theorem to study the existence and uniqueness of pseudo-almost periodic solutions to a class of abstract semilinear evolution equation in a Banach space. Our results complement a recent work due to Diagana (2008).

We consider pairs of non reflexive Banach spaces $$(E_0, E)$$ such that $$E_0$$ is defined in terms of a little-o condition and E is defined by the corresponding big-O condition. Under suitable assumptions on the pair $$(E_0, E)$$ there exists a reflexive and separable Banach space X (in which E is continuously embedded and dense) naturally associated to E which characterizes quantitatively weak compactness of bounded linear operators $$\begin{aligned} T: E_0 \rightarrow Z \end{aligned}$$ where Z is an arbitrary Banach space. Pairs include (VMO, BMO), where BMO is the space of John-Nirenberg, $$(B_0, B)$$ where B is a recently introduced space by Bourgain-Brezis-Mironescu ([6]) and some Orlicz pairs $$(L^{\psi }_0, L^{\psi })$$ where $$L^{\psi }_0$$ is the closure of $$L^\infty $$ in the Orlicz space $$L^{\psi }$$ , Marcinkiewicz pairs $$(L^{q, \infty }_0, L^{q, \infty })$$ where $$L^{q, \infty }_0$$ is the closure of $$L^\infty $$ in the Marcinkiewicz weak– $$L^q$$ denoted by $$L^{q, \infty }$$ . More generally, Banach function spaces are considered. The main results are duality formulas of the type 1 $$\begin{aligned} E_0^{**}&\simeq E\qquad \text {isometrically}\end{aligned}$$ 2 $$\begin{aligned}E^{*}&\simeq E_0^*\oplus _1 E_0^\perp\end{aligned}$$ and distance formulas.

Convergence of composite iterative methods for finding zeros of accretive operators

Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

In this chapter we introduce several most important notions and constructions concerning topological groups and operations with them. In particular, we develop the technique of prenorms on groups, describe in detail the construction of the Raĭkov completion of a topological group, prove a theorem on embeddings of topological groups into groups of isometries in the topology of pointwise convergence. We also introduce the class of locally compact topological groups and provide the reader with the first basic facts about groups in this class (deeper results in this direction will be presented in Chapter 9). Then we define and study the important classes of !-narrowtopological groups and of precompact (equivalently, totally bounded) topological groups. We finish this chapter with the Hartman{Mycielski construction of an embedding of an arbitrary topological group into a connected, locally pathwise connected topological group.