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Authors
Huaxin Lin
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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.
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We study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. We develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. We complement our generic results with the detailed study of many important special cases. In particular we study crossed products of tensor algebras, triangular AF algebras and various associated C*-algebras. We make contributions to the study of C*-envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. We also answer questions from the pertinent literature.
Starting from a discrete $C^*$-dynamical system $(mathfrak{A}, theta, omega_o)$, we define and study most of the main ergodic properties of the crossed product $C^*$-dynamical system $(mathfrak{A}rtimes_alphamathbb{Z}, Phi_{theta, u},om_ocirc E)$, $E:mathfrak{A}rtimes_alphamathbb{Z}rightarrowga$ being the canonical conditional expectation of $mathfrak{A}rtimes_alphamathbb{Z}$ onto $mathfrak{A}$, provided $ainaut(ga)$ commute with the $*$-automorphism $th$ up tu a unitary $uinga$. Here, $Phi_{theta, u}inaut(mathfrak{A}rtimes_alphamathbb{Z})$ can be considered as the fully noncommutative generalisation of the celebrated skew-product defined by H. Anzai for the product of two tori in the classical case.
A C*-dynamical system is said to have the ideal separation property if every ideal in the corresponding crossed product arises from an invariant ideal in the C*-algebra. In this paper we characterize this property for unital C*-dynamical systems over discrete groups. To every C*-dynamical system we associate a twisted partial C*-dynamical system that encodes much of the structure of the action. This system can often be untwisted, for example when the algebra is commutative, or when the algebra is prime and a certain specific subgroup has vanishing Mackey obstruction. In this case, we obtain relatively simple necessary and sufficient conditions for the ideal separation property. A key idea is a notion of noncommutative boundary for a C*-dynamical system that generalizes Furstenbergs notion of topological boundary for a group.
We consider a twisted noncommutative join procedure for unital $C^*$-algebras which admit actions by a compact abelian group $G$ and its discrete abelian dual $Gamma$, so that we may investigate an analogue of Baum-Dabrowski-Hajac noncommutative Borsuk-Ulam theory in the twisted setting. Namely, under what conditions is it guaranteed that an equivariant map $phi$ from a unital $C^*$-algebra $A$ to the twisted join of $A$ and $C^*(Gamma)$ cannot exist? This pursuit is motivated by the twisted analogues of even spheres, which admit the same $K_0$ groups as even spheres and have an analogous Borsuk-Ulam theorem that is detected by $K_0$, despite the fact that the objects are not themselves deformations of a sphere. We find multiple sufficient conditions for twisted Borsuk-Ulam theorems to hold, one of which is the addition of another equivariance condition on $phi$ that corresponds to the choice of twist. However, we also find multiple examples of equivariant maps $phi$ that exist even under fairly restrictive assumptions. Finally, we consider an extension of unital contractibility (in the sense of Dabrowski-Hajac-Neshveyev) modulo $k$.
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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