Discrete group actions

We complete classification of discrete abelian or finite group actions on injective type III1 factors up to cocycle conjugacy. We also give a proof for Connes’ characterization of the Ker (mod) and Cnt(M) for an injective factor M of type III. §0 Introduction. The purpose of this paper is to give a proof of Connes’ announcement on approximately inner automorphisms and centrally trivial automorphisms of an injective 1980 Mathematics Subject Classification (1985 Revision). 46L40.

It is shown that any semilattice of bicompact G-extensions can be realized as a semilattice for a pseudocompact phase space with an action of a discrete group. The following examples are given: lattices of bicompact G-extensions which cannot be lattices of bicompact extensions of any Tikhonov space and a pseudocompact G-space with a discrete acting group whose semilattice of bicompact G-extensions has a countable number of minimal elements.

Let m be a Radon measure on a Hausdorff topological space X.Corresponding to three kinds of outer measures, three kinds of m-negligible sets are considered.The main theorem states that in a metacompact space X each locally m-negligible set is m-negligible.For a Radon measure in a HausdorfT topological space X we distinguish three kinds of negligible sets corresponding to three kinds of outer measures: strictly negligible, negligible and locally negligible sets, respectively.It is shown that two classes of negligible sets coincide iff the corresponding outer measures coincide.Further, each locally negligible set is strictly negligible if X is Lindelf.We give two examples that both in locally compact separable spaces and in locally compact metrizable spaces a negligible set need not be strictly negligible.It is an unsolved problem whether a locally negligible set is always negligible, or equivalently, whether the corresponding outer measures are always the same.In this latter form the problem is due to Schwartz [5, p. 17].The main purpose of this note is to prove that in a metacompact space, and hence in a paracompact space, each locally negligible set is negligible.It will turn out that this is even true for compact inner regular Borel measures which are only locally a-finite.At least in this slightly more general situation it can be shown that the assumption " metacompact" cannot be replaced by "locally compact".By the compact inner regularity of Radon measures the necessity to distinguish several kinds of negligible sets arises with the infinity of the measures.That is, for finite, compact inner regular Borel measures the three kinds of negligible sets coincide.In the case of finite Borel measures (not necessarily compact inner regular) there are at most two kinds of negligible sets.The class of topological spaces in which these two kinds coincide for each finite or each finite regular Borel measure was considered by Gardner in [3] (see also [4]).Throughout this paper X is a Hausdorff topological space.(X), \}(X) and Si(X) denote, respectively, the families of all open, closed and compact subsets of X.The Borel a-algebra is denoted by %5(X).A Borel measure in X, i.e. a nonnegative, countably additive set function on 93 (X), is called locally finite if each point in X has a neighbourhood of finite measure.

In reminiscence of Ptak's open mapping theorem, a topological space satisfying the open mapping theorem is called a B r space.This paper is devoted to the study of sums and products of B r spaces in the category of topological spaces.We prove that, in general, sums and products of even two B r spaces need no longer be B r .On the other hand, for any B r space E the sum E θ E is again a B r space.Moreover, since Cech complete spaces are known to be B r , we ask whether a sum E θ F is B r provided that E is Cech complete and F is B r .It turns out that, at least in the framework of complete regularity, the answer to this question is in the positive if and only if F is a Baire space.Introduction.The notion of B r spaces has first been introduced by T. Husain in the categories of locally convex vector spaces (see [HuJ) and topological groups (see [Hu 2 ]).Originally, it goes back to V. Ptak's open mapping theorem (see [Kό], p. 35 ff).A Hausdorff topological space E is called a B r space if every continuous, nearly open bijection / from E onto any Hausdorff space F is in fact open.For a survey of the classical theory of B r spaces we refer the reader to Kόthe's book [Kό], where the locally convex case is treated.The linear topological case is investigated in the lecture notes [AEK].B r groups are considered by several authors.See for instance [Hu 2 ], [Ba], [Gr], [Su].In a purely topological context, B r spaces have first been investigated by Weston in [We], although the term B r space is not used there.Translated into the B r terminology, Weston proved that every completely metrizable topological space is a B r space.His result has been generalized by Byczkowski and Pol in [BP], who proved that every Cech complete topological space is a B r space.In [No] we have further generalized this result proving that every Hausdorff, semi-regular topological space densely containing some Cech complete sub space is in fact a B r space.We just mention another generalization of Byczkowski and Pols' result into a somewhat different direction by Wilhelm (cf.[Wi]).In the present paper we examine the invariance of the class of B r spaces under topological sums and products.It turns out that, in general, the sum of even two B r spaces need not be a B r space.A counterexample is given in §2.In §1 we obtain a positive result stating that the sum E θ E

We consider dual frames generated by actions of countable discrete groups on a Hilbert space. Module frames in a class of modules over a group algebra are shown to coincide with a class of ordinary frames in a representation of the group. This has applications to shift-invariant spaces and wavelet theory. One of the main findings in this paper is that whenever a shift-invariant subspace in $L^2(\mathbb {R}^n)$ has compactly supported dual frame generators, then it also has compactly supported bi-orthogonal generators. The crucial part in the proof is a theorem by Swan that states that every finitely generated projective module over the Laurent polynomials in $n$ variables is free.

In this paper we introduce (∈ − 2)-ball centered at each point in 2-metric topological space (X,d). Theorems on the normal, regular and Hausdorff topological spaces in 2-metrizable topological space are presented. We show that every metrizable topological spaces are coarser than 2-metrizable topological space, and then we conclude that each manifold is a 2-metric topological space.

For a non-elementary discrete isometry group G of divergence type acting on a proper geodesic delta -hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of G . As applications of this result, we have: (1) under a minor assumption, such a discrete group G admits no proper conjugation, that is, if the conjugate of G is contained in G , then it coincides with G ; (2) the critical exponent of any non-elementary normal subgroup of G is strictly greater than the half of that for G .

This thesis is dedicated to studying the theory of entropy and its relation to the Kolmogorov complexity. Originating in physics, the notion of entropy was introduced to mathematics by C. E. Shannon as a way of measuring the rate at which information is coming from a data source. There are, however, a few different ways of telling how much information there is: an alternative approach to quantifying the amount of information is the Kolmogorov complexity, which was proposed by A. N. Kolmogorov. The Shannon entropy is the key ingredient in the definition of the Kolmogorov-Sinai entropy of a measure-preserving systems. Roughly speaking, the Kolmogorov-Sinai entropy is the expected amount of information in `Shannon sense' that one obtains per unit of time by observing the evolution of a measure-preserving system. In topological dynamics, the topological entropy takes place of the Kolmogorov-Sinai entropy. For metrizable systems, the topological entropy measures the exponential growth rate of the number of distinguishable partial orbits of length n as n tends to infinity. Originally defined for Z-actions, the 'classical' theories of entropy were later extended to actions of amenable groups. We provide a necessary background on amenable groups, topological/measurepreserving dynamics and the entropy theory in Chapters 1, 2, 3 and 5. The main focus of this thesis is extending the following results. First of all, a common generalization of the topological and the Kolmogorov-Sinai entropy theories for Z-systems was suggested by G. Palm. We provide an abstract generalization of the work of Palm for actions of discrete amenable groups in the language of measurement functors in Chapter 6. Secondly, we investigate the connection of entropy and Kolmogorov complexity. Originally, the equality between the topological entropy and a certain quantity measuring maximal asymptotic Kolmogorov complexity of the trajectories was established by A. A. Brudno for subshifts over Z. Later, he proved the equality of the Kolmogorov-Sinai entropy and the asymptotic Kolmogorov complexity of almost every trajectory for ergodic subshifts over Z. We provide a generalization of these results as follows. Firstly, in Chapter 4 we give a background on computability and Kolmogorov complexity and, further, introduce computable Folner monotilings, which are central in our extensions of Brudno's results. We treat the 'first' and the 'second' theorems of Brudno in Chapter 7. The first theorem is generalized for subshifts over computable groups admitting computable Folner monotilings, while the second theorem is proved under the assertion of regularity of the monotiling, which we introduce in Chapter 7 as well.

We study actions of arithmetic groups on compact manifolds which preserve a connection.

We construct, for each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e. each of these spaces is a collection of quasifolds glued together in an suitable way. A quasifold is a space locally modelled on \({\mathbb{R}^{k}}\) modulo the action of a discrete, possibly infinite, group. The way strata are glued to each other also involves the action of an (infinite) discrete group. Each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the original polytope back. These spaces may be viewed as a natural generalization of symplectic toric varieties to the nonrational setting.

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

We prove a xed point theorem for a class of discrete group acting on man ifolds of nonpositive curvature by isometry These discrete groups include cocompact lattices in simply connected semisimple p adic groups of rank at least two and large p Hence it gives a geometric generalization of Margulis superrigidity theorem for the Archimedean representation of these groups Introduction Let N be a complete simply connected Riemannian manifold of non positive sectional curvature N has a natural compacti cation N N N by the sphere at in nity which is de ned as the equivalent classes of geodesic rays Any group action on N by isometry extends to an action on N De nition A group is said to have property F if any isometric action of on any complete simply connected manifold of nonpositive sectional curvature N has a xed point in N In term of representations of property F has the following inter pretation Let H be a simple noncompact Lie group with trivial center If has property F then any homomorphism H with Zariski dense image is precompact inH This is because the symmetric space as sociated with H has nonpositive curvature and the image being Zariski Received October and in revised form July

Some Results on Bundle Systems for a Countable Discrete Amenable Group Action

Hyperbolic spaces Hn [7] are models of an infinite cosmos with constant negative curvature. Moreover they provide universal covering spaces for compact finite manifolds M of nontrivial topology with the same metric and curvature. The double torus T∪T and the Weber-Seifert manifold are hyperbolic manifolds M covered by H 2, H 3 respectively. Both display closed geodesics with observable effects on the distribution of matter. The preimage and image points under any homotopy g ∈ π(M) when acting on Hn are identified on Μ. Therefore any closed geodesic corresponds to a homotopy g. It is shown that the closed geodesics on M associated with g can be compared and classified in length and relative direction by orbits under the continuous normalizers Ng , g ∈ π(M).

Given an action of a discrete group on a smooth compact manifold with a boundary, we consider a class of operators generated by pseudodifferential operators on and shift operators associated with the group action. For elliptic operators in this class, we obtain a classification up to stable homotopies and show that the group of stable homotopy classes of such problems is isomorphic to the -group of the crossed product of the algebra of continuous functions on the cotangent bundle over the interior of the manifold and the group acting on this algebra by automorphisms.