Frames generated by actions of countable discrete groups

We exhibit examples of actions of countable discrete groups on both simple and non‐simple nuclear stably finite C*‐algebras that are tracially amenable but not amenable. We furthermore obtain that, under the additional assumption of strict comparison, amenability is equivalent to tracial amenability plus the equivariant analogue of Matui–Sato's property (SI). By virtue of this equivalence, our construction yields the first known examples of actions on classifiable C*‐algebras that do not have equivariant a over show that such actions can be chosen to absorb the trivial action on the universal UHF algebra, thus proving that equivariant ‐stability does not in general imply equivariant property (SI).

<p style='text-indent:20px;'>In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula> action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, the centralizer of any minimal aperiodic <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-action on the Cantor set is a subgroup of the centralizer of a minimal <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula>-action.

On a central algorithm for calculation of the inverse of the harmonic oscillator in the spaces of orbits

We establish several classification results for compact extensions of tracial W^{*} -dynamical systems and for relatively independent joinings thereof for actions of arbitrary discrete groups. We use these results to answer a question of Austin, Eisner, and Tao and some questions raised by Duvenhage and King. Moreover, combining our results with an earlier classification of weakly mixing extensions by Popa, we can derive non-commutative Furstenberg–Zimmer type dichotomies on the L^{2} -level. Although in general an adequate generalization of the Furstenberg–Zimmer tower of intermediate compact extensions does not seem possible in the von Neumann algebraic framework, we show that there always exists a non-commutative analogue of the finer Host–Kra–Ziegler tower for any ergodic action of a countable abelian group.

In a paper entitled cocycles and spectra Helson and Par ry prove that for every T there exists a q~ such that V ~ has Lebesgue spectrum, moreover q~ can be chosen to be real, i.e., taking values + 1 and 1. The purpose of this paper is to extend this result to certain actions of countable groups which includes ergodic non-singular actions of countable abelian groups. We blend the method of Helson and Parry with the notions of weak equivalence and weak von Neumann transformations. In section 3 and 4 we discuss these results in connection with systems of imprimitivity. The problem of extending the result quoted above to countable groups was raised by H. Helson to one of us. It is a pleasure to acknowledge his interest and encouragement in this work.

Fixed points for bounded orbits in Hilbert spaces

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

This is a short review of recent constructions of new Calabi-Yau threefolds with small Hodge numbers and/or nontrivial fundamental group, which are of particular interest for model building in the context of heterotic string theory. The two main tools are topological transitions and taking quotients by actions of discrete groups. Both of these techniques can produce new manifolds from existing ones, and they have been used to bring many new specimens to the previously sparse corner of the Calabi-Yau zoo, where both Hodge numbers are small. Two new manifolds are also obtained here from hyperconifold transitions, including the first example with fundamental groupS3, the smallest non-Abelian group.

Our ongoing goal in this part of the text is to understand quotient spaces that locally look like (products of) hyperbolic spaces. In order to get off the ground, here we put the previous two chapters in a more general context, seeking to understand nice group actions on topological spaces and indicating how these fit in with more general notions in topology. Pathologies exist! Our goal in this chapter is to provide basic context (for further references see 34.5.4) before turning to the central case of interest: a discrete group acting properly on a locally compact, Hausdorff topological space.

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.

For a given Hilbert space $$\mathcal H$$ , consider the space of self-adjoint projections $$\mathcal P(\mathcal H)$$ . In this paper we study the differentiable structure of a canonical sphere bundle over $$\mathcal P(\mathcal H)$$ given by $$\begin{aligned} \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \Vert f\Vert =1\, \}. \end{aligned}$$ We establish the smooth action on $$\mathcal R$$ of the group of unitary operators of $$\mathcal H$$ , and it thereby turns out that the connected components of $$\mathcal R$$ are homogeneous spaces. Then we study the metric structure of $$\mathcal R$$ by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into $$\mathcal R$$ by the natural action of the unitary group. Then we study the restricted bundle $$\mathcal R_2^+$$ given by considering only the projections in the restricted Grassmannian, locally modeled by Hilbert–Schmidt operators. Therefore we endow $$\mathcal R_2^+$$ with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi–Civita connection of this metric and establish a Hopf–Rinow theorem for $$\mathcal R_2^+$$ , again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.

We introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context.

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.

Sofic groups were defined implicitly by Gromov and explicitly by Weiss. All residually finite groups (and hence all linear groups) are sofic. The purpose of this paper is to introduce, for every countable sofic group G G , a family of measure-conjugacy invariants for measure-preserving G G -actions on probability spaces. These invariants generalize Kolmogorov-Sinai entropy for actions of amenable groups. They are computed exactly for Bernoulli shifts over G G , leading to a complete classification of Bernoulli systems up to measure-conjugacy for many groups, including all countable linear groups. Recent rigidity results of Y. Kida and S. Popa are utilized to classify Bernoulli shifts over mapping class groups and property (T) groups up to orbit equivalence and von Neumann equivalence, respectively.