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

To study outer actions $\alpha$ of a group $G$ on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$, $0<\lambda<1$, we study first the cohomology group of a group with the unitary group of an abelian von Neumann algebra as a coefficient group and establish a technique to reduce the coefficient group to the torus $\mathsf T$ by the Shapiro mechanism based on the groupoid approach. We then show a functorial construction of outer actions of a countable discrete amenable group on an AFD factor of type $\mathrm{III}_\lambda$, sharpening the result in [17, §4]. The periodicity of the flow of weights on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$ allows us to introduce an equivariant commutative square directly related to the discrete core. But this makes it necessary to introduce an enlarged group $\mathrm{Aut}(\mathcal M)_{m}$ relative to the modulus homomorphism $m=\mod\colon \mathrm{Aut}(\mathcal M)\to \mathsf R/T'\mathsf Z$. We then discuss the reduced modified HJR-exact sequence, which allows us to describe the invariant of outer action $\alpha$ in a simpler form than the one for a general AFD factor: for example, the cohomology group $H_{m,s}^{out}(G,N,\mathsf T)$ of modular obstructions is a compact abelian group. Making use of these reductions, we prove the classification result of outer actions of $G$ on an AFD factor $\mathcal M$ of type $\mathrm{III}_{\lambda}$.

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.

KMS states of self-similar k-graph C*-algebras

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.

Let G be a countable exact discrete group. G has the strong invariant approximation property(SIAP) if and only if C � u(G,S) G = C � � (G) ⊗ S for any Hilbert space H and closed subspace S ⊆ H. We shall use results of Haagerup and Kraus on the approximation property (AP) to investigate some permanence properties of the SIAP for discrete groups. This can be done most efficiently for exact groups. In this paper we describe that the stability properties of the SIAP property pass to semi direct products, and extensions for discrete exact groups.

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.

We consider a countable discrete group Γ \Gamma acting ergodically on a standard Borel space S with quasi-invariant measure μ \mu . Let π \pi be a unitary representation of Γ \Gamma on L 2 ( S , d μ , H ) {L^2}(S,d\mu ,\mathcal {H}) "nicely" related with S. We prove that if Γ \Gamma acts amenably on S then π \pi is weakly contained in the regular representation.

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We prove that for actions of a discrete countable residuallyfinite amenable group on a compact metric space with specification property, periodic measures are dense in theset of invariant measures. We also prove that certain expansiveactions of a countable discrete group by automorphisms of compact abelian groups have specification property.

Let ( X , Γ ) (X, \Gamma ) be a free and minimal topological dynamical system, where X X is a separable compact Hausdorff space and Γ \Gamma is a countable infinite discrete amenable group. It is shown that if ( X , Γ ) (X, \Gamma ) has the Uniform Rokhlin Property (URP) and Cuntz comparison of open sets (COS), then m d i m ( X , Γ ) = 0 \mathrm {mdim}(X, \Gamma )=0 implies that ( C ( X ) ⋊ Γ ) ⊗ Z ≅ C ( X ) ⋊ Γ (\mathrm {C}(X) \rtimes \Gamma )\otimes \mathcal Z \cong \mathrm {C}(X) \rtimes \Gamma , where m d i m \mathrm {mdim} is the mean dimension of ( X , Γ ) (X, \Gamma ) , Z \mathcal Z is the Jiang-Su algebra, and C ( X ) ⋊ Γ \mathrm {C}(X) \rtimes \Gamma is the transformation group C*-algebra of ( X , Γ ) (X, \Gamma ) . In particular, in this case, m d i m ( X , Γ ) = 0 \mathrm {mdim}(X, \Gamma )=0 implies that the C*-algebra C ( X ) ⋊ Γ \mathrm {C}(X) \rtimes \Gamma is classified by the Elliott invariant.

Given an infinite countable discrete amenable group Γ \Gamma , we construct explicitly sharply weak mixing nonsingular Poisson Γ \Gamma -actions of each Krieger’s type: I I I λ III_\lambda , for λ ∈ [ 0 , 1 ] \lambda \in [0,1] , and I I ∞ II_\infty . The result is new even for Γ = Z \Gamma =\mathbb {Z} . As these Poisson suspension actions are over very special dissipative base, we obtain also new examples of sharply weak mixing nonsingular Bernoulli Γ \Gamma -actions and infinite direct product of finite type systems of each possible Krieger’s type.

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsion. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C ∗ \mathrm {C}^{*} -algebra of a countable discrete quantum group \mathbbl {\Gamma } implies that \mathbbl {\Gamma } is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition.

We consider a crossed product of a unital simple separable nuclear stably finite $\cal Z$-stable $C^*$-algebra $A$ by a strongly outer cocycle action of a discrete countable amenable group $\Gamma$. Under the assumption that $A$ has finitely many extremal tracial states and $\Gamma$ is elementary amenable, we show that the twisted crossed product $C^*$-algebra is $\cal Z$-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on $\cal Z$ are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite $C^*$-algebras.

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).

In this paper we consider a bootstrap class $$\mathfrak {C}$$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on $$\mathrm {C}^*$$ -algebras. This class includes all torsion-free abelian groups, poly- $$\mathbb {Z}$$ -groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $$\Gamma \in \mathfrak {C}$$ and any strongly self-absorbing $$\mathrm {C}^*$$ -algebra $$\mathcal {D}$$ : In fact we establish more general relative versions of these two results for actions of amenable groups that have a predetermined quotient in the class $$\mathfrak {C}$$ . For the monotracial case, the proof comprises an application of Matui–Sato’s equivariant property (SI) as a key method.