Applications of amenable semigroups in operator theory

The paper deals with continuous homomorphisms $S \ni s \mapsto T_s \in L(E)$ of amenable semigroups $S$ into the algebra $L(E)$ of all bounded linear operators on a Banach space $E$. For a closed linear subspace $F$ of $E$, sufficient conditions are given under which there exists a projection $P \in L(E)$ onto $F$ that commutes with all $T_s$. And when $E$ is a Hilbert space, sufficient conditions are given for the existence of an invertible operator $R \in L(E)$ such that all $R T_s R^{-1}$ are isometries. Also certain results on extending intertwining operators, renorming as well as on operators on hereditarily indecomposable Banach spaces are offered.Comment: 16 page

Topological groups: where to from here?

This is an account of one man's view of the current perspective of theory of topological groups. We survey some recent developments which are, from our viewpoint, indicative of the future directions, concentrating on actions of topological groups on compacta, embeddings of topological groups, free topological groups, and `massive' groups (such as groups of homeomorphisms of compacta and groups of isometries of various metric spaces).Comment: To appear in Proceedings of the 14-th Summer Topology Conference (NY, Aug. 1999), final versio

Definitions of almost periodicity

Various versions of the classical definitions of (one- and twosided) almost periodicity for functions on groups with values in a uniform space are formulated and their equivalence is shown.Comment: 18 page

Weakly almost periodic topologies, idempotents and ideals

Let (G,tau_G) be a topological group. We establish relationships between weakly almost periodic topologies on G coarser than tau_G, central idempotents in the weakly almost periodic compactification G^W, and certain ideals in the algebra of weakly almost periodic functions W(G). We gain decompositions of weakly almost periodic representations, generalizing many from the literature. We look at the role of pre-locally compact topologies, unitarizable topologies, and extend or decompositions to Fourier-Stieltjes algebras B(G).Comment: 31 pages, V2 has some corrected mistakes and typos, and a new example added to Section 6.

Compact operator semigroups applied to dynamical systems

In this paper we develop a systematic theory of compact operator semigroups on locally convex vector spaces. In particular we prove new and generalized versions of the mean ergodic theorem and apply them to different notions of mean ergodicity appearing in topological dynamics

Specification properties and thermodynamical properties of semigroup actions

In the present paper we study the thermodynamical properties of finitely generated continuous subgroup actions. We address a notion of topological entropy and pressure functions that does not depend on the growth rate of the semigroup and introduce strong and orbital specification properties, under which, the semigroup actions have positive topological entropy and all points are entropy points. Moreover, we study the convergence and Lipschitz regularity of the pressure function and obtain relations between topological entropy and exponential growth rate of periodic points in the context of semigroups of expanding maps. The specification properties for semigroup actions and the corresponding one for its generators and the action of push-forward maps is also discussed.Comment: 26 pages; Final version to appear in Journal of Mathematical Physic

Uniform families of ergodic operator nets

We study mean ergodicity in amenable operator semigroups and establish the connection to the convergence of strong and weak ergodic nets. We then use these results in order to show the convergence of uniform families of ergodic nets that appear in topological Wiener-Wintner theorems.Comment: 14 page

A Remark on the structure of symmetric quantum dynamical semigroups on von Neumann algebras

We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup with norm-bounded generator on a von Neumann algebra equipped with a faithful semifinite trace. For von Neumann algebras with abelian commutant (i.e. type I von Neumann algebras), we give a necessary and sufficient algebraic condition for the generator of such a semigroup to be written as a sum of square of self-adjoint derivations of the von Neumann algebra. This generalizes some of the results obtained by Albeverio, H(phi)egh-Krohn and Olsen [Alb] for the special case of the finite dimensional matrix algebras. We also study similar questions for a class of quantum dynamical semigroups with unbounded generators.Comment: accepted in Infinite Dimensional Analysis, Quantum Probability and Related Toplics (World Scientific

Compactifications, Hartman functions and (weak) almost periodicity

In this paper we investigate Hartman functions on a topological group $G$. Recall that $(\iota, C)$ is a group compactification of $G$ if $C$ is a compact group, $\iota: G\to C$ is a continuous group homomorphism and $\iota(G)$ is dense in $C$. A complex-valued bounded function $f$ on $G$ is a Hartman function if there exists a group compactification $(\iota, C)$ and a complex-valued bounded function $F$ on $C$ such that $f=F\circ\iota$ and $F$ is Riemann integrable, i.e. the set of discontinuities of $F$ is a null set with respect to the Haar measure. In particular we answer the question how large a compactification for a given group $G$ and a Hartman function $f$ must be, to admit a Riemann integrable representation of $f$. In order to give a systematic presentation which is self-contained to a reasonable extent, we include several separate sections on the underlying concepts such as finitely additive measures on Boolean set algebras, means on algebras of functions, integration on compact spaces, compactifications of groups and semigroups, the Riemann integral on abstract spaces, invariance of measures and means, continuous extensions of transformations and operations to compactifications, etc.Comment: 64 page

Variants on the Berz sublinearity theorem

We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function -- results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.Comment: Dedication: To Roy O. Davies on his 90-th birthday. Keywords: Dependent Choices, Homomorphism extension, Subadditivity, Sublinearity, Steinhaus-Weil property, Thinning, Quantifier weakenin