Representations of groups on Banach spaces

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.

1. Let (X, T, 7r) be a topological transformation group, where X is a nontrivial Hausdorff space and T is a topological group which leaves an end point e of X fixed. In [3], Wallace proved that if T is cyclic and X is a locally connected continuum, T has a fixed point other than e. Then in [4], Wallace asked the following question: If X is a peano continuum and T is compact or abelian, then does T have a fixed point other than e? In [5] Wang showed that if T is compact, and X is arcwise connected, then T has a fixed point other than e. Then Chu [1 ] showed that T has infinitely many fixed points. Chu began the study of the abelian case in [2]. In this paper we show by example that X may be a peano continuum and T may be a countably generated abelian group which has e for its only fixed point, ?2. Thus in general the answer to Wallace's question in the abelian case is no. However, if T is a generative group, and X is compact and arcwise connected, then T has a fixed point other than e, ?3. We also show by example that T may be a finitely generated nonabelian group which has e for its only fixed point, ?4.

The topological properties of "weak mixing" and "strong mixing" for topological transformation groups have been introduced and investigated and many analogous results have been derived. Let us adopt the following definitions and notations. We will restrict our attention to topological transformation groups (X, T) with compact metrizable phase space X. We call a topological transformation group ergodic if the only closed proper invariant subset is nowhere dense or, equivalently, there exists a point in X the orbit-closure of which is the whole space X. Such a point is called a transitive point. We call a topological transformation group weakly mixing if (X, T)x (X, T) is ergodic and strongly mixing if for any two open sets U and V, there is a compact subset K of T such that t ~ T ~ K implies Ut n V # ;J. Let ~'(X) be the set of all bounded complex-valued functions on X which are continuous on a co-meager set. (We identify functions which coincide on a co-meager set.) Let us define a cascade to be a topological transformation group with the discrete additive group of integers as phase group. In this situation, it is also denoted by (X, ~b), with ~b a homeomorphism of X. If Y is an invariant subset of X, then (Y, ~b) is also a cascade and we say that (Y, ~b) is a subcascade of (X, ~b). Let us consider the following properties of a topological transformation group (X, T):

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.

Let 5C be a complex Hubert Space, J3(3C) the ring of bounded operators on 3C, an abelian symmetric subring of B(3Z) containing the identity which is closed in the weak operator topology, E the commutant of E, and suppose E has a cyclic vector £o which we normalize so that |£o| = 1 . Diximier [ l] has shown that (respect. Ei), as a Banach space, is the dual of the Banach space R (respect. Ri) of all linear forms on (respect. E\) that are continuous in the ultra-strong topology of (respect. Ei). In this note we show that every TCzR is also continuous in the weak operator topology of E, from which it follows that a linear functional T on is continuous in either the weak, ultraweak, strong, or ultrastrong topologies if and only if it is continuous in all four simultaneously. In the process, we obtain an integral representation for such T, which we later use in a theorem on centrally reducible positive functionals on E%. We denote the maximal ideal space of by M, and for A, B, • • • £ , we denote the corresponding Gel'fand transforms by a, 6, • • • . Then A—>a is an isometric isomorphism from £ onto C(M). Consequently, every bounded linear functional on has the form

Introduction. This article is based upon a principle which is so standard that it is almost a cliche: The first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation. To be more specific, we shall describe some applications of this to the symmetry questions of topological transformation groups. Given a topological space X, let Homeo(^) denote the set of self-homeomorphisms of X. This is a group under composition of mappings. If G is an arbitrary group, then a group action of G on I is a homomorphism Homeo( X). Frequently we wish to impose some weak assumptions on <p. For example, if G is a topological group, then we might want cp to have suitable continuity properties. The usual assumption is that the map

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.

Chapter 42 - Recent results and open questions relating Chu duality and Bohr compactifications of locally compact groups

Let H be a hypergroup with left Haar measure and let L1(H) be the complex Lebesgue space associated with it. Let L∞(H) be the set of all locally measurable functions that are bounded except on a locally null set, modulo functions that are zero locally a.e. It is a standard device to embed L∞(H) into ℬ(L1(H),L∞(H)). We denote the strong operator topology and the weak operator topology on L∞(H) by τso and τwo. Unlike the uniform norm and weak topologies on L∞(H), they depend essentially on the hypergroup structure of H. We derive that the τso-topology is always different from the weak∗-topology whenever H is infinite. We can conclude that for a compact hypergroup H, L1(H) is the dual of (L∞(H),τso). The properties of τso and τwo are then studied further and we pay attention to the τwo-almost periodic elements of L∞(H). Finally we give some further results about bounded linear operators which are τso-τso-continuous.

Let /" be a group of finite homological type, X a finite dimensional, free /'-complex such that H.(X, Z) is finitely generated.We proved that H,(X//', Z)is finitely generated, and x(X//')-X(1-')" 2: (X), where Z (/') is the Euler characteristic of a group Key words: Topological transformation groups; Euler characteristics of groups.

Metrizability of topological groups having weak topologies with respect to good covers

We describe continuous linear operators acting in a countable inductive limit of weighted Frchet spaces of entire functions of several complex variables and commuting in these spaces with systems of partial differentiation and translation operators. Under the made assumptions, the commutants of the systems of differentiation and translation operators coincide. They consist of convolution operators defined by an arbitrary continuous linear functional on . At that, we do not assume that the set of the polynomials is dense in . In the space topological dual to , we introduce the natural multiplication. Under this multiplication, the algebra is isomorphic to the aforementioned commutant with the usual multiplication, which is the composition of the operators. This isomorphism is also topological if is equipped by the weak topology, while the commutant is equipped by the weak operator topology. This implies that the set of the polynomials of the differentiation operators is dense in the commutant with topology of pointwise convergence. We also study the possibility of representing an operator in the commutant as an infinite order differential operator with constant coefficients. We prove the immediate continuity of linear operators commuting with all differentiation operators in a weighted (LF)-space of entire functions isomorphic via Fourier-Laplace transform to the space of infinitely differentiable functions compactly supported in a real multi-dimensional space.

We prove that the topology of the additive group of the Banach space $c_0$ is not induced by weakly almost periodic functions or, what is the same, that this group cannot be represented as a group of isometries of a reflexive Banach space. We show, in con

Let G be a reflexive topological group, and Gits group of characters, endowed with the compact open topology. We prove that the evaluation mapping from Gx G into the torus T is continuous if and only if G is locally compact. This is an analogue of a well-known theorem of Arens on admissible topologies on C(X) . DEFINITIONS AND REMARKS Let X, Y be topological spaces, and let Z be a subset of yX . A topology on Z is said to be admissible if the evaluation mapping from the product Z x X into Y, defined by w(f, x) = f(x), is continuous. Let (S, V) = {f E Z; f(S) C V}. The family {(S, V)}, where S runs over the collection of all compact subsets of X and V runs over a basis of open sets in Y, is a subbase for the compact open topology on Z. An admissible topology on Z must be finer than the compact open topology [8]. A result of Arens states that the existence of a coarsest admissible topology for the class of real continuous functions on a completely regular space X is equivalent to X being locally compact [1]. In this paper we are interested in reflexive topological groups. Answering in the negative a question of Megrelishvili [7], we will see that the evaluation map for those groups need not be continuous. In fact, as specified in the theorem, the continuity of the evaluation characterizes locally compact Hausdorff abelian groups among reflexive groups. We prove this result by means of convergence spaces. For an account of theory of convergence spaces the reader is referred to [3, 4]. We only give here needed definitions. Let G be a Hausdorff topological abelian group, and let FG be the set of all continuous homomorphisms from G into the unidimensional torus T. If addition is defined pointwise in FG, it becomes an abelian group. The continuous Received by the editors October 5, 1993 and, in revised form, April 25, 1994; the contents of this paper were presented to the First Ibero-American Conference on Topology and its Applications in Benicassim, Spain, on March 28-30, 1995. 1991 Mathematics Subject Classification. Primary 22A05.