We give the first examples of (non-amenable group) amenable actions on stably finite simple C*-algebras. More precisely, we give such actions for any countable group in an explicit way. The main ingredients of our construction are the full Fock space of the regular representation and a trace-scaling automorphism.
We show, based on previous results, that two separable simple stably projectionless amenable ${\cal Z}$-stable $C^*$-algebras which satisfy the UCT are isomorphic if and only if they have the same Elliott invariant.
A countable group is C*-simple if its reduced C*-algebra is simple. It is well known that C*-simplicity implies that the amenable radical of the group must be trivial. We show that the converse does not hold by constructing explicit counter-examples. We additionally prove that every countable group embeds into a countable group with trivial amenable radical and that is not C*-simple.
We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes") with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-sta…
We introduce an Ulam-type stability condition for positive definite maps defined on a countable group and prove that this condition characterizes amenability.
We associate a cohomological invariant to each outer action of a group on a factor, and classify them by the invariant in the case that the group is a countable discrete amenable group and the factor is appoximately finite dimensional. The invariant defined for the group Out(M)=Aut(M)/Int(M) is called the intrinsic modular obstruction. The invariant for an outer action alpha is given as the pull back of the intrinsic modular obstruction, which is called the modular obstruction of alpha and denoted by Ob_m(alpha). This is the first part of the theory and presents general theory. In the case tha…
We construct a locally compact groupoid with the properties in the title. Our example is based closely on constructions used by Higson, Lafforgue, and Skandalis in their work on counterexamples to the Baum-Connes conjecture. It is a bundle of countable groups over the one point compactification of the natural numbers, and is Hausdorff, second countable and étale.
We present a stable uniqueness theorem for non-unital C*-algebras. Generalized tracial rank one is defined for stably projectionless simple C*-algebras. Let $A$ and $B$ be two stably projectionless separable simple amenable C*-algebras with $gTR(A)\le 1$and $gTR(B)\le 1.$ Suppose also that $KK(A, D)=KK(B,D)=\{0\}$ for all C*-algebras $D.$ Then $A\cong B$ if and only if they have the same tracial cones with scales. We also show that every separable simple C*-algebra, $A$ with finite nuclear dimension which satisfies the UCT with non-zero traces must have $gTR(A)\le 1$ if $K_0(A)$ is torsion. In…
Following Elliott's earlier work, we show that the Elliott invariant of any finite separable simple $C^*$-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable abelian group which unifies the unital and nonunital, as well as stably projectionless cases. We also show that, for any given such invariant set, there is a finite separable simple $C^*$-algebra, whose Elliott invariant is the given set, a refinement of the range theorem of Elliott in the stable case. In the stably projectionless case, modified model $C^*$-algeb…
We present a classification theorem for a class of unital simple separable amenable ${\cal Z}$-stable $C^*$-algebras by the Elliott invariant. This class of simple $C^*$-algebras exhausts all possible Elliott invariant for unital stably finite simple separable amenable ${\cal Z}$-stable $C^*$-algebras. Moreover, it contains all unital simple separable amenable $C^*$-alegbras which satisfy the UCT and have finite rational tracial rank.
We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear dimension and with UCT such that every trace is quasi-diagonal has the property that $A\otimes Q$ has generalized tracial rank at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott.
We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G. Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first l2-Betti number of G with that of the stabilizer subgroups. An analogous relationship is also shown to hold for cost. This relationship becomes even more pronounced for transitive amenable actions, leading to a simple criterion for vanishing of the first l2-Betti number and triviality of cost. Moreover, …
The goals of this paper are twofold. First, we generalize the result of Gaboriau and Lyons [GL07] to the setting of von Neumann's problem for equivalence relations, proving that for any non-amenable ergodic probability measure preserving (pmp) equivalence relation $\mathcal{R}$, the Bernoulli extension over a non-atomic base space $(K, κ)$ contains the orbit equivalence relation of a free ergodic pmp action of $\mathbb{F}_2$. Moreover, we provide conditions which imply that this holds for any non-trivial probability space $K$. Second, we use this result to prove that any non-amenable unimodula…
We say that an action of a countable discrete inverse semigroup on a locally compact Hausdorff space is amenable if its groupoid of germs is amenable in the sense of Anantharaman-Delaroche and Renault. We then show that for a given inverse semigroup $S$, the action of $S$ on its spectrum is amenable if and only if every action of $S$ is amenable.
We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. Some structural question concerning these simple C*-algebras are studied. The paper also serves as a technical support for the classification of separable stably projectionless simple amenable Jiang-Su stable C*-algebras.
In this last article of the series on outer actions of a countable dicrete amenable group on AFD factors, we analyze outer actions of a countable discrete free abelian group on an AFD factor of type $\text{III}_λ$, $0<λ< 1$, and compute outer conjugacy invariants. As a byproduct, we discover the asymmetrization technique for coboundary condition on a $\Bbb T$-valued cocycle of a torsion free abelian group, which might have been known by the group cohomologists. As the asymmetrization technique gives us a very handy criteria for coboundaries, we present it here in detail in the second section.
Kida and Tucker-Drob recently extended the notion of inner amenability from countable groups to discrete p.m.p. groupoids. In this article, we show that inner amenable groupoids have "fixed priced 1" in the sense that every principal extension of an inner amenable groupoid has cost 1. This simultaneously generalizes and unifies two well known results on cost from the literature, namely, (1) a theorem of Kechris stating that every ergodic p.m.p. equivalence relation admitting a nontrivial asymptotically central sequence in its full group has cost 1, and (2) a theorem of Tucker-Drob stating that…
We study actions of countable discrete amenable groups on unital separable simple nuclear Z-absorbing C*-algebras. Under a certain assumption on tracial states, which is automatically satisfied in the case of a unique tracial state, the crossed product is shown to absorb the Jiang-Su algebra Z tensorially.
We completely characterize connected Lie groups all of whose countable subgroups are weakly amenable. We also provide a characterization of connected semisimple Lie groups that are weakly amenable. Finally, we show that a connected Lie group is weakly amenable if the group is weakly amenable as a discrete group.
It is shown that if A is a separable, exact C*-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then A admits a trace-preserving embedding into a simple, unital AF-algebra with unique trace. Modulo the UCT, this provides an abstract characterization of C*-subalgebras of simple, unital AF-algebras. As a consequence, for a countable, discrete, amenable group G acting on a second countable, locally compact, Hausdorff space X, C_0(X) \rtimes_r G embeds into a simple, unital AF-algebra if, and only if, X admits a faithful, invariant, Borel, probabi…
A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras. Moreover, it contains all unital simple separable amenable C*-algebras which satisfy the UCT and have finite rational tracial rank
We consider a crossed product of a unital simple separable nuclear stably finite Z-stable C*-algebra A by a strongly outer cocycle action of a discrete countable amenable group Γ. Under the assumption that A has finitely many extremal tracial states and Γis elementary amenable, we show that the twisted crossed product C*-algebra is Z-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on 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 establish that a second countable locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable. We give some examples and applications.
We prove a cocycle superrigidity theorem for a large class of coinduced actions. In particular, if $Λ$ is a subgroup of a countable group $Γ$, we consider a probability measure preserving action $Λ\curvearrowright X_0$ and let $Γ\curvearrowright X$ be the coinduced action. Assume either that $Γ$ has property (T) or that $Λ$ is amenable and $Γ$ is a product of non-amenable groups. Using Popa's deformation/rigidity theory we prove $Γ\curvearrowright X$ is $\mathcal U_{fin}$-cocycle superrigid, that is any cocycle for this action to a $\mathcal U_{fin}$ (e.g. countable) group $\mathcal V$ is coho…
We show that any stationary symmteric $α$-stable ($SαS$) random field indexed by a countable amenable group $G$ is weakly mixing if and only if it is generated by a null action, extending works of Samorodnitsky and Wang-Roy-Stoev for abelian groups to all amenable groups. This enables us to improve significantly the domain of a recently discovered connection to von Neumann algebras. We also establish ergodicity of stationary $SαS$ fields associated with boundary and double boundary actions of a hyperbolic group $G$, where the boundary is equipped with either the Patterson-Sullivan or the hitti…
We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag-Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, $\widetilde{A}_2$-groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and…
We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact second countable group is exact if and only if it admits a topologically amenable action on a compact Hausdorff space. This answers an open question by Anantharaman-Delaroche.
We prove an inverse theorem for the Gowers $U^2$-norm for maps $G\to\mathcal M$ from an countable, discrete, amenable group $G$ into a von Neumann algebra $\mathcal M$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert\cdot\Vert$. We use this result to prove a stability result for unitary-valued $\varepsilon$-representations $G\to\mathcal U(\mathcal M)$ with respect to $\Vert\cdot \Vert$.
In this short note we construct two countable, infinite conjugacy class groups which admit free, ergodic, probability measure preserving orbit equivalent actions, but whose group von Neumann algebras are not (stably) isomorphic.
We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of p-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1-expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclic-by-finite group on X, it is shown that the entropy of the a…
For a countable amenable group Γand an element f in the integral group ring ZΓbeing invertible in the group von Neumann algebra of Γ, we show that the entropy of the shift action of Γon the Pontryagin dual of the quotient of ZΓby its left ideal generated by f is the logarithm of the Fuglede-Kadison determinant of f. For the proof, we establish an \ell^p-version of Rufus Bowen's definition of topological entropy, addition formulas for group extensions of countable amenable group actions, and an approximation formula for the Fuglede-Kadison determinant of f in terms of the determinants of pertur…
We prove that the unitary equivalence classes of extensions of C*_r(G) by any sigma-unital stable C"-algebra, taken modulo extensions which split via an asymptotic homomorphism, form a group which can be calculated from the universal coefficient theorem of KK-theory when G is a free product of a countable collection of amenable groups.
We show that, for countable sofic groups, a Bernoulli action with infinite entropy base has infinite entropy with respect to every sofic approximation sequence. This builds on the work of Lewis Bowen in the case of finite entropy base and completes the computation of measure entropy for Bernoulli actions over countable sofic groups. One consequence is that such a Bernoulli action fails to have a generating countable partition with finite entropy if the base has infinite entropy, which in the amenable case is well known and in the case that the acting group contains the free group on two genera…
We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $Γ$ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $Γ$-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $Γ_n$-fixed points, where $(Γ_n, n\ge1)$ is a decreasing sequence of finite index…
Given a minimal action $α$ of a countable group on the Cantor set, we show that the alternating full group $\mathsf{A}(α)$ is non-amenable if and only if the topological full group $\mathsf{F}(α)$ is $C^*$-simple. This implies, for instance, that the Elek-Monod example of non-amenable topological full group coming from a Cantor minimal $\mathbb{Z}^2$-system is $C^*$-simple.
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 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 $Γ\in\mathfrak C$ and any strongly self-absorbing C*-algebra $\mathcal D$: (1) There is a unique strongly …
Let $G$ be a countable discrete amenable group, ${\cal M}$ a McDuff factor von Neumann algebra, and $A$ a separable nuclear weakly dense C$^*$-subalgebra of ${\cal M}$. We show that if two centrally free actions of $G$ on ${\cal M}$ differ up to approximately inner automorphisms then they are outer conjugate by an approximately inner automorphism, in the operator norm topology, which makes $A$ invariant. In addition, when $A$ is unital, simple, and with a unique tracial state and $α$ is an automorphism of $A$ we also show that the aperiodicity of $α$ on the von Neumann algebra is equivalent to…
For each group $G$ having an infinite normal subgroup with the relative property (T) (for instance $G = H \times K$ where $H$ is infinite with property (T) and $K$ is arbitrary), and any countable abelian group $Λ$ we construct free ergodic measure preserving actions $σ_Λ$ of $G$ on the probability space such that the 1'st cohomology group of $σ_Λ$, $H^1(σ_Λ)$, is equal to Char$(G) \times Λ$. We deduce that $G$ has uncountably many non stably orbit equivalent actions. We also calculate 1-cohomology groups and show existence of ``many'' non stably orbit equivalent actions for free products of g…
We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is…
We undertake a comprehensive study of measure equivalence between general locally compact, second countable groups, providing operator algebraic and ergodic theoretic reformulations, and complete the classification of amenable groups within this class up to measure equivalence.
We show that a large class of i.c.c., countable, discrete groups satisfying a weak negative curvature condition are not inner amenable. By recent work of Hull and Osin, our result recovers that mapping class groups and Out(F_n) are not inner amenable. We also show that the group-measure space constructions associated to free, strongly ergodic p.m.p. actions of such groups do not have property Gamma of Murray and von Neumann.
Given a simple, acyclic dimension group $G_{0}$ and countable, torsion-free, abelian group $G_{1}$, we construct a minimal, amenable, étale equivalence relation $R$ on a Cantor set whose associated groupoid $C^{*}$-algebra, $C^{*}(R)$, is tracially AF, and hence classifiable in the Elliott classification scheme for simple, amenable, separable $C^{*}$-algebras, and with $K_{*}(C^{*}(R)) \cong(G_{0}, G_{1})$.
Building on work of Popa, Ioana, and Epstein--Törnquist, we show that, for every nonamenable countable discrete group $Γ$, the relations of conjugacy, orbit equivalence, and von Neumann equivalence of free ergodic (or weak mixing) measure preserving actions of $Γ$ on the standard atomless probability space are not Borel, thus answering questions of Kechris. This is an optimal and definitive result, which establishes a neat dichotomy with the amenable case, since any two free ergodic actions of an amenable group on the standard atomless probability space are orbit equivalent by classical result…
We prove W$^*$-superrigidity for a large class of coinduced actions. We prove that if $Σ$ is an amenable almost-malnormal subgroup of an infinite conjugagy class (icc) property (T) countable group $Γ$, the coinduced action $Γ\curvearrowright X$ from an arbitrary probability measure preserving action $Σ\curvearrowright X_0$ is W$^*$-superrigid. We also prove a similar statement if $Γ$ is an icc non-amenable group which is measure equivalent to a product of two infinite groups. In particular, we obtain that any Bernoulli action of such a group $Γ$ is W$^*$-superrigid.
We study the ideal structure of reduced crossed product of topological dynamical systems of a countable discrete group. More concretely, for a compact Hausdorff space $X$ with an action of a countable discrete group $Γ$, we consider the absence of a non-zero ideals in the reduced crossed product $C(X) \rtimes_r Γ$ which has a zero intersection with $C(X)$. We characterize this condition by a property for amenable subgroups of the stabilizer subgroups of $X$ in terms of the Chabauty space of $Γ$. This generalizes Kennedy's algebraic characterization of the simplicity for a reduced group $\mathr…
The generalized Effros-Hahn conjecture for groupoid C*-algebras says that, if G is amenable, then every primitive ideal of the groupoid C*-algebra C*(G) is induced from a stability group. We prove that the conjecture is valid for all second countable amenable locally compact Hausdorff groupoids. Our results are a sharpening of previous work of Jean Renault and depend significantly on his results.
