Let A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element $σ$ of the parametrized Kasparov group KK_X(A,B) is invertible if and only if all its fiberwise components $σ_x\in KK(A(x),B(x))$ are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra O_2. Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.
We introduce notions of continuous orbit equivalence and strong (respective, weak) continuous orbit equivalence for automorphism systems of étale equivalence relations, and characterize them in terms of the semi-direct product groupoids, as well as their reduced groupoid $C^*$-algebras and the associated $C^*$-automorphism systems of group actions or coactions on them. In particular, we study topological rigidity of expansive automorphism actions on compact (connected) metrizable groups
We introduce a notion of uniformly continuous orbit equivalence as a subequivalence relation of continuous orbit equivalence of one-sided topological Markov shifts. It is described in terms of gauge actions on the associated Cuntz-Krieger algebras and continuous full groups of the Markov shifts.
For continuously orbit equivalent one-sided topological Markov shifts $(X_A,σ_A)$ and $(X_B,σ_B)$, their eventually periodic points and cocycle functions are studied. As a result we directly construct an isomorphism between their ordered cohomology groups $(\bar{H}^A, \bar{H}^A_+)$ and $(\bar{H}^B, \bar{H}^B_+)$. We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of the ordered cohomology, so that the the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invar…
We will introduce a notion of strongly continuous orbit equivalence in one-sided topological Markov shifts. Strongly continuous orbit equivalence yields a topological conjugacy between their two-sided topological Markov shifts $(\bar{X}_A, \barσ_A)$ and $(\bar{X}_B, \barσ_B)$. We prove that one-sided topological Markov shifts $(X_A, σ_A)$ and $(X_B, σ_B)$ are strongly continuous orbit equivalent if and only if there exists an isomorphism bewteen the Cuntz-Krieger algebras ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$ preserving their maximal commutative $C^*$-subalgebras $C(X_A)$ and $C(X_B)$ and gi…
We consider $C^*$-algebras constructed from compact group actions on complex vector bundles $E\to X$ endowed with a Hermitian metric. An action of $G$ by isometries on $E\to X$ induces an action on the $C^*$-correspondence $Γ(E)$ over $C(X)$ consisting of continuous sections, and on the associated Cuntz-Pimsner algebra $\mathcal O_E$, so we can study the crossed product $\mathcal O_E\rtimes G$. If the action is free and rank $E=n$, then we prove that $\mathcal O_E\rtimes G$ is Morita-Rieffel equivalent to a field of Cuntz algebras $\mathcal O_n$ over the orbit space $X/G$. If the action is fib…
We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.
We will study torus actions on Cuntz--Krieger algebras trivially acting on its canonical maximal abelian $C^*$-subalgebras from the view points of continuous orbit equivalence of one-sided topological Markov shifts and flow equivalence of two-sided topological Markov shifts.
In this paper, we consider semigroup actions of discrete countable semigroups on compact spaces by surjective local homeomorphisms. We introduce notions of continuous one-sided orbit equivalence and continuous orbit equivalence for semigroup actions, and characterize them in terms of the corresponding semi-groupoids and transformation groupoids respectively. Finally, we consider the case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of group actions.
Drimbe and Vaes proved an orbit equivalence superrigidity theorem for left-right wreath product actions in the measurable setting. We establish the counterpart result in the topological setting for continuous orbit equivalence. This gives us minimal, topologically free actions that are continuous orbit equivalence superrigid. One main ingredient for the proof is to show continuous cocycle superrigidity for certain generalized full shifts, extending our previous result with Chung.
Let $A$ be an $N\times N$ irreducible matrix with entries in $\{0,1\}$. We present an easy way to find an $(N+3)\times (N+3)$ irreducible matrix $\bar{A}$ with entries in $\{0,1\}$ such that their Cuntz--Krieger algebras ${\mathcal{O}}_A$ and ${\mathcal{O}}_{\bar{A}}$ are isomorphic and $ \det(1 -A) = - \det(1-\bar{A}). $ As a consequence, we know that two Cuntz--Krieger algebras ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$ are isomorphic if and only if the one-sided topological Markov shift $(X_A, σ_A)$ is continuously orbit equivalent to $(X_B, σ_B)$ or $(X_{\bar{B}}, σ_{\bar{B}}).$
We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterise continuous orbit equivalence in terms of isomorphisms of C*-crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and Feldman-Moore in the measurable setting. Secondly, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this …
We obtain a weak homotopy equivalence type result between two topological groups associated with a Kirchberg algebra: the unitary group of the continuous asymptotic centralizer and the loop group of the automorphism group of the stabilization. This result plays a crucial role in our subsequent work on the classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras. As a special case, we show that the $K$-groups of the continuous asymptotic centralizer are isomorphic to the $KK$-groups of the Kirchberg algebra.
We extend ultragraph shift spaces and the realization of ultragraph C*-algebras as partial crossed products to include ultragraphs with sinks (under a mild condition, called (RFUM2), which allow us to dismiss the use of filters) and we describe the associated transformation groupoid. Using these characterizations we study continuous orbit equivalence of ultragraph shift spaces (via groupoids) and KMS and ground states (via partial crossed products).
We study KMS states for gauge actions with potential functions on Cuntz--Krieger algebras whose underlying one-sided topological Markov shifts are continuous orbit equivalent. As a result, we have a certain relationship between topological entropy of continuous orbit equivalent one-sided topological Markov shifts.
We study asymptotic continuous orbit equivalence of Smale spaces. We prove that two irreducible Smale spaces are flip conjugate if and only if there exists a periodic point preserving homeomorphism giving an asymptotic continuous orbit equivalence between them. We introduce a notion of asymptotic topological conjugacy and asymptotic flip conjugacy in Smale spaces and characterize them in terms of the associated Ruelle algebras with dual actioons. We finally characterize the flip conjugacy classes of irreducible two-sided topological Markov shifts in terms of the associated Ruelle algebras with…
We consider fiberwise singly generated Fell-bundles over etale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo-Martin-Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of traces on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a des…
A $λ$-graph system $\frak L$ is a labeled Bratteli diagram with shift operation. It is a generalized notion of finite labeled graph and presents a subshifts. We will study continuous orbit equivalence of one-sided subshifts and topological conjugacy of two-sided subshifts from the view points of groupoids and $C^*$-algebras constructed from $λ$-graph systems.
We prove that the automorphisms of any separable C*-algebra that does not have continuous trace are not classifiable by countable structures up to unitary equivalence. This implies a dichotomy for the Borel complexity of the relation of unitary equivalence of automorphisms of a separable unital C*-algebra: Such relation is either smooth or not even classifiable by countable structures.
In the first part of the paper, we introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their asymptotic Ruelle algebras with their dual actions. In the second part, we introduce a groupoid $C^*$-algebra which is an extended version of the asymptotic Ruelle algebra from a Smale space and study the extended Ruelle algebras from the view points of Cuntz--Krieger algebras. As a result, the asymptotic Ruelle algebra is realized as a fixed point algebra of the extended Ruelle algebra under certain circle action.
We define Schatten classes of adjointable operators on Hilbert modules over abelian $C^*$-algebras. Many key features carry over from the Hilbert space case. In particular, the Schatten classes form two-sided ideals of compact operators and are equipped with a Banach norm and a $C^*$-valued trace with the expected properties. For trivial Hilbert bundles, we show that our Schatten-class operators can be identified bijectively with Schatten-norm-continuous maps from the base space into the Schatten classes on the Hilbert space fiber, with the fiberwise trace. As applications, we introduce the $C…
In this article, we generalize to the case of regular locally compact quantum groups, two important results concerning actions of compact quantum groups. Let $G_1$ and $G_2$ be two monoidally equivalent regular locally compact quantum groups in the sense of De Commer. We introduce an induction procedure and we build an equivalence of the categories ${A}^{G_1}$ and ${A}^{G_2}$ consisting of continuous actions of $G_1$ and $G_2$ on $C^*$-algebras. As an application of this result, we derive a canonical equivalence of the categories ${KK}^{G_1}$ and ${KK}^{G_2}$. We introduce and investigate a no…
We prove that for any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space, they are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group with certain other virtually cyclic groups, e.g. the direct product of the integer group with any non-abelian finite simple group.
Let A,B be square irreducible matrices with entries in {0,1}. We will show that if the one-sided topological Markov shifts (X_A,σ_A) and (X_B,σ_B) are continuously orbit equivalent, then the two-sided topological Markov shifts (\bar X_A,\barσ_A) and (\bar X_B,\barσ_B) are flow equivalent, and hence det(id-A)=det(id-B). As a result, the one-sided topological Markov shifts (X_A,σ_A) and (X_B,σ_B) are continuously orbit equivalent if and only if the Cuntz-Krieger algebras O_A and O_B are isomorphic and det(id-A)=det(id-B).
To every one-sided shift space $\mathsf{X}$ we associate a cover $\tilde{\mathsf{X}}$, a groupoid $\mathcal{G}_{\mathsf{X}}$ and a $\mathrm{C^*}$-algebra $\mathcal{O}_{\mathsf{X}}$. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer preserving) continuous orbit equivalence between $\mathsf{X}$ and $\mathsf{Y}$ in terms of isomorphism of $\mathcal{G}_{\mathsf{X}}$ and $\mathcal{G}_{\mathsf{Y}}$, and diagonal preserving $^*$-isomorphism of $\mathcal{O}_{\mathsf{X}}$ and $\mathcal{O}_{\mathsf{Y}}$. We also characterize two-sided conjugacy and flow equivalence of the associate…
We characterize Lipschitz morphisms between quantum compact metric spaces as those *-morphisms which preserve the domain of certain noncommutative analogues of Lipschitz seminorms, namely lower semi-continuous Lip-norms. As a corollary, lower semi-continuous Lip-norms with a shared domain are in fact equivalent. We then note that when a family of lower semi-continuous Lip-norms are uniformly equivalent, then they give rise to totally bounded classes of quantum compact metric spaces, and we apply this observation to several examples of perturbations of quantum metric spaces. We also construct t…
We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As …
We show that for any locally compact Hausdorff space $Y$ with finite covering dimension and for any continuous flow $\mathbb{R} \curvearrowright Y$, the resulting crossed product $C^*$-algebra $C_0(Y) \rtimes \mathbb{R}$ has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. As an application, we obtain bounds for the nuclear dimension of $C^*$-algebras associated to one-dimensional orientable foliations. This result is analogous to the one we obtained earlier for non-free actions of $\mathbb{Z}$. Some novel tec…
We introduce the notion of strong Morita equivalence for group actions on locally C*-algebras and prove that the crossed products associated with two strongly Morita equivalent continuous inverse limit actions of a locally compact group G on the locally C*-algebras A and B are strongly Morita equivalent. This generalizes a result of F. Combes, Proc. London Math. Soc. 49(1984) and R. E. Curto, P.S. Muhly, D. P. Williams, Proc. Amer. Soc. 90(1984).
Let A and B be normal matrices with coefficients that are continuous complex-valued functions on a topological space X that has the homotopy type of a CW complex, and suppose these matrices have the same distinct eigenvalues at each point of X. We use obstruction theory to establish a necessary and sufficient condition for A and B to be unitarily equivalent. We also determine bounds on the number of possible unitary equivalence classes in terms of cohomological invariants of X.
A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of groups including the lamplighter group and all virtually solvable groups having finite Prüfer rank. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this…
