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Register a new userMinimal dynamical systems on connected odd dimensional spaces
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Huaxin Lin
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Let $beta: S^{2n+1}to S^{2n+1}$ be a minimal homeomorphism ($nge 1$). We show that the crossed product $C(S^{2n+1})rtimes_{beta} Z$ has rational tracial rank at most one. More generally, let $Omega$ be a connected compact metric space with finite covering dimension and with $H^1(Omega, Z)={0}.$ Suppose that $K_i(C(Omega))=Zoplus G_i$ for some finite abelian group $G_i,$ $i=0,1.$ Let $beta: OmegatoOmega$ be a minimal homeomorphism. We also show that $A=C(Omega)rtimes_{beta}Z$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This was done by studying the minimal homeomorphisms on $Xtimes Omega,$ where $X$ is the Cantor set.
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Let $X$ be an infinite compact metric space with finite covering dimension and let $alpha, beta : Xto X$ be two minimal homeomorphisms. We prove that the crossed product $C^*$-algebras $C(X)rtimes_alphaZ$ and $C(X)rtimes_beltaZ$ are isomorphic if and only if they have isomorphic Elliott invariant. In a more general setting, we show that if $X$ is an infinite compact metric space and if $alpha: Xto X$ is a minimal homeomorphism such that $(X, alpha)$ has mean dimension zero, then the tensor product of the crossed product with a UHF-algebra of infinite type has generalized tracial rank at most one. This implies that the crossed product is in a classifiable class of amenable simple $C^*$-algebras.
We investigate some ergodic and spectral properties of general (discrete) $C^*$-dynamical systems $({mathfrak A},Phi)$ made of a unital $C^*$-algebra and a multiplicative, identity-preserving $*$-map $Phi:{mathfrak A}to{mathfrak A}$, particularising the situation when $({mathfrak A},Phi)$ enjoys the property of unique ergodicity with respect to the fixed-point subalgebra. For $C^*$-dynamical systems enjoying or not the strong ergodic property mentioned above, we provide conditions on $lambda$ in the unit circle ${zin{mathbb C}mid |z|=1}$ and the corresponding eigenspace ${mathfrak A}_lambdasubset{mathfrak A}$ for which the sequence of Cesaro averages $left(frac1{n}sum_{k=0}^{n-1}lambda^{-k}Phi^kright)_{n>0}$, converges point-wise in norm. We also describe some pivotal examples coming from quantum probability, to which the obtained results can be applied.
We provide some properties and characterizations of homologically $UV^n$-maps and $lc^n_G$-spaces. We show that there is a parallel between recently introduced by Cauty algebraic $ANR$s and homologically $lc^n_G$-metric spaces, and this parallel is similar to the parallel between ordinary $ANR$s and $LC^n$-metric spaces. We also show that there is a similarity between the properties of $LC^n$-spaces and $lc^n_G$-spaces. Some open questions are raised.
A compact space $X$ is said to be minimal if there exists a map $f:Xto X$ such that the forward orbit of any point is dense in $X$. We consider rigid minimal spaces, motivated by recent results of Downarowicz, Snoha, and Tywoniuk [J. Dyn. Diff. Eq., 2016] on spaces with cyclic group of homeomorphisms generated by a minimal homeomorphism, and results of the first author, Clark and Oprocha [ Adv. Math., 2018] on spaces in which the square of every homeomorphism is a power of the same minimal homeomorphism. We show that the two classes do not coincide, which gives rise to a new class of spaces that admit minimal homeomorphisms, but no minimal maps. We modify the latter class of examples to show for the first time the existence of minimal spaces with degenerate homeomorphism groups. Finally, we give a method of constructing decomposable compact and connected spaces with cyclic group of homeomorphisms, generated by a minimal homeomorphism, answering a question in Downarowicz et al.
In this paper, we accomplish two objectives. Firstly, we extend and improve some results in the theory of (semi-)strongly self-absorbing C*-dynamical systems, which was introduced and studied in previous work. In particular, this concerns the theory when restricted to the case where all the semi-strongly self-absorbing actions are assumed to be unitarily regular, which is a mild technical condition. The central result in the first part is a strengthened version of the equivariant McDuff-type theorem, where equivariant tensorial absorption can be achieved with respect to so-called very strong cocycle conjugacy. Secondly, we establish completely new results within the theory. This mainly concerns how equivariantly $cal Z$-stable absorption can be reduced to equivariantly UHF-stable absorption with respect to a given semi-strongly self-absorbing action. Combining these abstract results with known uniqueness theorems due to Matui and Izumi-Matui, we obtain the following main result. If $G$ is a torsion-free abelian group and $cal D$ is one of the known strongly self-absorbing C*-algebras, then strongly outer $G$-actions on $cal D$ are unique up to (very strong) cocycle conjugacy. This is new even for $mathbb{Z}^3$-actions on the Jiang-Su algebra.
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