Michael Megrelishvili

Fragmentability and Continuity of Semigroup Actions

fragmentability in the sense of Jayne and Rogers and its natural uniform generalizations play a major role in this paper. Our applications show that problems concerning the continuity of induced actions have satisfactory solutions for Asplund Banach spaces X (without additional restrictions, if S is a topological group) and, moreover, for a new locally convex version of Asplund spaces introduced in the paper. The starting point of this concept was the characterization of Asplund spaces due to Namoika and Phelps in terms of fragmentability.

Globalization of confluent partial actions on topological and metric spaces

Abstract We generalize Exels notion of partial group action to monoids. For partial monoid actions that can be defined by means of suitably well-behaved systems of generators and relations, we employ classical rewriting theory in order to describe the universal induced global action on an extended set. This universal action can be lifted to the setting of topological spaces and continuous maps, as well as to that of metric spaces and non-expansive maps. Well-known constructions such as Shimrats homogeneous extension are special cases of this construction. We investigate various properties of the arising spaces in relation to the original space; in particular, we prove embedding theorems and preservation properties concerning separation axioms and dimension. These results imply that every normal (metric) space can be embedded into a normal (metrically) ultrahomogeneous space of the same dimension and cardinality.

Topological transformation groups: Selected topics

Publisher Summary This chapter discusses selected topics related to topological transformation groups. In the discussion presented, all topological spaces are Tychonoff. A topological transformation group, or a G -space is a triple ( G , X , π ), wherein the continuous action of a topological group G on a topological space X is π : G × X → X , π ( g , x ) := gx . Supposing that G acts on X 1 and on X 2 , a continuous map f : X 1 → X 2 is a G-map (or, an equivariant map) if f(gx ) = gf ( x ) for every ( g , x ) ∈ G × X 1 . The Banach algebra of all continuous real valued bounded functions, on a topological space X , is denoted by C ( X ). If ( G , X , π) be a G -space, it induces the action G × C ( X ) → C ( X ), with ( gf )( x ) = f ( g −1 x ). A function f ∈ C ( X ) is said to be right uniformly continuous, or also π-uniform, if the map G → C ( X ), g ↦gf is norm continuous. Concepts related to equivariant compactifications and equivariant normality are also elaborated. Details of universal actions are also provided in the chapter.

Group representations and construction of minimal topological groups

Abstract For every continuous biadditive mapping ω we construct a topological group M(ω) and establish its minimality under natural restrictions. Using the evaluation mapping G × G ∗ → T of Pontryagin-van Kampen duality and the canonical duality E × E ∗ → R for a normed space E, we obtain some new results in the theory of minimal groups. In particular, it is shown that every locally compact Abelian group is a group retract of a minimal locally compact group. Every Abelian topological group is a quotient of a perfectly minimal group.

Minimality Conditions in Topological Groups

This is a survey on the recent progress in minimal topological groups, with a particular emphasis on constructions leading to non-abelian minimal groups, as semidirect products, generalized Heisenberg groups and other groups naturally arising in Analysis and Geometry. A special attention is paid to several generalizations of minimality (as local minimality, relative minimality and co-minimality), the relations of minimality to (dis)connectedness and to various level of compactness and completeness.

Uniformities and uniformly continuous functions on locally connected groups

We show that the left and the right uniformities on a locally connected topological group G coincide if and only if every left uniformly continuous real-valued function on G is right uniformly continuous.

Representations of Dynamical Systems on Banach Spaces

We review recent results concerning dynamical systems and their representations on Banach spaces. As the enveloping (or Ellis) semigroup, which is associated to every dynamical system, plays a crucial role in this investigations we also survey the new developments in the theory of these semigroups, complementing the review article [49]. We then discuss some applications of these dynamical results to topological groups and Banach spaces.

Banach Representations and Affine Compactifications of Dynamical Systems

To every Banach space V we associate a compact right topological affine semigroup ℰ(V ). We show that a separable Banach space V is Asplund if and only if \(\mathcal{E}(V )\) is metrizable, and it is Rosenthal (i.e., it does not contain an isomorphic copy of l 1) if and only if \(\mathcal{E}(V )\) is a Rosenthal compactum. We study representations of compact right topological semigroups in \(\mathcal{E}(V )\). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily nonsensitive) dynamical systems, respectively. As an application we obtain a generalization of a theorem of R. Ellis. A main theme of our investigation is the relationship between the enveloping semigroup of a dynamical system X and the enveloping semigroup of its various affine compactifications Q(X). When the two coincide we say that the affine compactification Q(X) is E-compatible. This is a refinement of the notion of injectivity. We show that distal non-equicontinuous systems do not admit any E-compatible compactification. We present several new examples of non-injective dynamical systems and examine the relationship between injectivity and E-compatibility.

Circularly ordered dynamical systems

We study topological properties of circularly ordered dynamical systems and prove that every such system is representable on a Rosenthal Banach space, hence, is also tame. We derive some consequences for topological groups. We show that several Sturmian like symbolic