We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Consider a stably finite simple unital C*-algebra, and an automorphism of it which has the tracial Rokhlin property. Suppose the algebra has real rank zero and stable rank one, and suppose that the order on projections over the algebra is determined by traces. Then the crossed product by the automorphism also has these three properties.
We also present examples of C*-algebras and automorphisms which satisfy the above assumptions, but such that crossed product algebras do not have tracial rank zero.
Tracial Rokhlin property was introduced by Phillips to prove various structure theorems for crossed product. But it is defined for actions on simple C*-algebras only. This paper consists of two major parts: In section 2 and 3, we study the permanence properties and give a complete classification of tracial Rokhlin property for product-type actions; In section 4 and 5, we introduce the weak tracial Rokhlin property for actions on non-simple C*-algebras. We prove that when the action has the weak tracial Rokhlin property and the crossed product is simple, the properties on $A$ of having tracial …
We define "tracial" analogs of the Rokhlin property for actions of finite groups, approximate representability of actions of finite abelian groups, and of approximate innerness. We prove four analogs of related "nontracial" results. First, the crossed product of an infinite dimensional simple separable unital C*-algebra with tracial rank zero by an action of a finite group with the tracial Rokhlin property again has tracial rank zero. Second, an outer action of a finite abelian group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property if and only if i…
We show by construction that when $G$ is an elementary amenable group and $A$ is a unital simple nuclear and tracially approximately divisible $C^*$-algebra, there exists an action $ω$ of $G$ on $A$ with the tracial Rokhlin property in the sense of Matui and Sato. In particular, group actions with this Matui-Sato tracial Rokhlin property always exist for unital simple separable nuclear $C^*$-algebras with tracial rank at most one. If $A$ is simple with rational tracial rank at most one, then the crossed product $A\rtimes_ωG$ is also simple with rational tracial rank at most one.
We define a "tracial" analog of the Rokhlin property for actions of second countable compact groups on infinite dimensional simple separable unital C*-algebras. We prove that fixed point algebras under such actions (and, in the appropriate cases, crossed products by such actions) preserve simplicity, Property (SP), tracial rank zero, tracial rank at most one, the Popa property, tracial Jiang-Su stability, Jiang-Su stability when the algebra is nuclear, infiniteness, and pure infiniteness. We also show that the radius of comparison of the fixed point algebra is no larger than that of the origin…
We define the tracial Rokhlin property for actions of finite cyclic groups on stably finite simple unital C*-algebras. We prove that when the algebra is in addition simple and has tracial rank zero, then the crossed product again has tracial rank zero. Under a kind of weak approximate innerness assumption and one other technical condition, we prove that if the action has the the tracial Rokhlin property and the crossed product has tracial rank zero, then the original algebra has tracial rank zero. We give examples showing how the tracial Rokhlin property differs from the Rokhlin property of Iz…
We introduce the tracial quasi-Rokhlin property for an automorphism alpha of a unital C*-algebra A, which is not assumed to be simple. We show that under suitable hypotheses, the associated crossed product C*-algebra C*(Z,A,alpha) is simple, and there is a bijection between the space of tracial states on C*(Z,A,alpha)$ and the alpha-invariant tracial states on A. We show that, for a minimal dynamical system (X,h) and a simple, separable, unital C*-algebra A, the automorphism beta which extends the action of h on C(X) has the tracial quasi-Rokhlin property, and hence that C*(Z,C(X,A),beta) has …
Tracial Rokhlin property was introduced by Chris Phillips to study the structure of crossed product of actions on simple C*-algebras. It was originally defined for actions of finite groups and group of integers. Matui and Sato generalized it to actions of amenable groups. In this paper, we give a further generalization of Matui and Sato's definition. We shall show that many known results about tracial Rokhlin property could be generalized to actions of amenable groups under this definition. We also show that some natural examples has the (weak) tracial Rokhlin property.
We prove several results of the following general form: automorphisms of (or actions of ${\mathbb{Z}}^d$ on) certain kinds of simple separable unital C*-algebras $A$ which have a suitable version of the Rokhlin property are generic among all automorphisms (or actions), or in a suitable class of automorphisms. That is, the ones with the version of the Rokhlin property contain a dense $G_δ$-subset of the set of all such automorphisms (or actions). Specifically, we prove the following. If $A$ is stable under tensoring with the Jiang-Su algebra $Z,$ and has tracial rank zero, then automorphisms wi…
We give examples of actions of Z/2Z on AF algebras and AT algebras which demonstrate the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability and tracial approximate representability. Specific results include the following. We determine exactly when a product type action of Z/2Z on a UHF algebra has the tracial Rokhlin property; in particular, unlike for the strict Rokhlin property, every UHF algebra admits such an action. We prove that Blackadar's action of Z/2Z on the 2^{\infty} UHF algebra, whose crossed produc…
In this paper we introduce an analog of the tracial Rokhlin property, called the {\emph {projection free tracial Rokhlin property}}, for $C^*$-algebras which may not have any nontrivial projections. Using this we show that if $A$ is an infinite dimensional stably finite simple unital $C^*$-algebra with stable rank one, with strict comparison of positive elements, with only finitely many extreme tracial states, and with the property that every 2-quasi-trace is a trace, and if $α$ is an action of a finite group $G$ with the projection free tracial Rokhlin property, then the crossed product $C^*(…
Let $A$ be a unital separable simple \CA with $\tr(A)\le 1$ and $α$ be an automorphism. We show that if $α$ satisfies the tracially cyclic Rokhlin property then $\tr(A\rtimes_α\Z)\le 1.$ We also show that whenever $A$ has a unique tracial state and $α^m$ is uniformly outer for each $m (\not= 0)$ and $α^r$ is approximately inner for some $r>0,$ $α$ satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear \CA s, we use the above result to prove a conjecture of Kishimoto: if $A$ is a unital simple $A{\mathbb T}$-algebra of real rank zero and $α\in \Aut(A)$ w…
Let $A$ be a stably finite simple unital $C^*$-algebra and suppose $α$ is an action of a finite group $G$ with the tracial Rokhlin property. Suppose further $A$ has real rank zero and the order on projections over $A$ is determined by traces. Then the crossed product $C^*$-algebra $C^*(G,A, α)$ also has real rank zero and order on projections over $A$ is determined by traces. Moreover, if $A$ also has stable rank one, then $C^*(G,A, α)$ also has stable rank one.
Let A be a unital AF-algebra (simple or non-simple) and let αbe an automorphism of A. Suppose that αhas certain Rokhlin property and A is α-simple. Suppose also that there is an integer J\geq1 such that α^{J}_{*0}=id_{K_{0}(A)}, we show that A\rtimes_α\mathbb{Z} has tracial rank zero.
Let $A$ be a unital simple $A\T$-algebra of real rank zero. Given an isomorphism $γ_1: K_1(A)\to K_1(A),$ we show that there is an automorphism $\af: A\to A$ such that $\af_{*1}=γ_1$ which has the tracial Rokhlin property. Consequently, the crossed product $A\rtimes_{\af}\Z$ is a simple unital AH-algebra with real rank zero. We also show that automorphism with Rokhlin property can be constructed from minimal homeomorphisms on a connected compact metric space.
We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of C*-algebras, including: $D$-absorbing C*-algebras, where $D$ is a strongly self-absorbing C*-algebra, stable C*-algebras, C*-algebras with finite …
We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necessarily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction to invariant hereditary C*-algebras, minimal tensor products, and direct limits of actions. Some of these results are new even in the unital case and answer open questions asked by N. C. Phillips in full generality. We present several examples of finite group actions with the weak tracial Rokhlin property on simple stably projectionless C*-algebras. We prove …
We introduce and study a Rokhlin-type property for actions of finite groups on (not necessarily unital) C*-algebras. We show that the corresponding crossed product C*-algebras can be locally approximated by C*-algebras that are stably isomorphic to closed two-sided ideals of the given algebra. We then use this result to prove that several classes of C*-algebras are closed under crossed products by finite group actions with this Rokhlin property.
We define a tracial analogue of the sequentially split $*$-homomorphism between $C^*$-algebras of Barlak and Szabó and show that several important approximation properties related to the classification theory of $C^*$-algebras pass from the target algebra to the domain algebra. Then we show that the tracial Rokhlin property of the finite group $G$ action on a $C^*$-algebra $A$ gives rise to a tracial version of sequentially split $*$-homomorphism from $A\rtimes_αG$ to $M_{|G|}(A)$ and the tracial Rokhlin property of an inclusion $C^*$-algebras $A\subset P$ with a conditional expectation $E:A \…
Let $\mathcal{A}$ be the class of unital separable simple amenable $C$*-algebras $A$ which satisfy the Universal Coefficient Theorem for which $A\otimes M_{\texttt{P}}$ has tracial rank zero for some supernatural number $\texttt{p}$ of infinite type. Let $A\in \mathcal{A}$ and let $α$ be an automorphism of $A.$ Suppose that $α$ has the tracial Rokhlin property. Suppose also that there is an integer $J\geq 1$ such that $[α^J]=[\mbox{id}_A]$ in $KL(A,A)$, we show that $A\rtimes_α\mathbb{Z}\in \mathcal{A}.$
Following the results known in the case of a finite abelian group action on $C\sp*$-algebras we prove the following two theorems; 1. an inclusion $P\subset A$ of (Watatani) index-finite type has the Rokhlin property (is approximately representable) if and only if the dual inclusion is approximately representable (has the Rokhlin property). 2. an inclusion $P\subset A$ of (Watatani) index-finite type has the tracial Rokhlin property (is tracially approximately representable) if and only if the dual inclusion is tracially approximately representable (has the tracial Rokhlin property). Moreover, …
We establish four results concerning connections between actions on separable C*-algebras with Rokhlin-type properties and absorption of the Jiang-Su algebra Z. For actions of residually finite groups or of the reals which have finite Rokhlin dimension with commuting towers, we show that if the action of any nontrivial group element is approximately inner then the C*-algebra acted upon is Z-stable. Without the assumption on approximate innerness, we show that the crossed product has good divisibility properties under mild assumptions. We also establish an analogous result for the generalized t…
We study a general Kishimoto's problem for automorphisms on simple C*-algebras with tracial rank zero. Let $A$ be a unital separable simple C*-algebra with tracial rank zero and let $α$ be an automorphism. Under the assumption that $α$ has certain Rokhlin property, we present a proof that $A\rtimes_α\Z$ has tracial rank zero. We also show that if the induced map $α_{*0}$ on $K_0(A)$ fixes a "dense" subgroup of $K_0(A)$ then the tracial Rokhlin property implies a stronger Rokhlin property. Consequently, the induced crossed product C*-algebras have tracial rank zero.
We introduce the notion of the weak tracial approximate representability of a discrete group action on a unital $C^*$-algebra which could have no projections like the Jiang-Su algebra $\mathcal{Z}$. Then we show a duality between the weak tracial Rokhlin property and the weak tracial approximate representability. More precisely, when $G$ is a finite abelian group and $α:G\curvearrowright A$ is a group action on a unital simple infinite dimensional $C^*$-algebra, we prove that 1. $α$ has the weak tracial Rokhlin property if and only if $\hatα$ has the weak tracial approximate representability. …
Let $A$ be a simple, exact, separable, unital $C^*$-algebra and let $α\colon G \rightarrow Aut(A)$ be an action of a finite group $G$ with the weak tracial Rokhlin property. We show that every trace on $A \rtimes_α G$ is quasidiagonal provided that all traces on $A$ are quasidiagonal. As an application, we study the behavior of finite decomposition rank under taking crossed products by finite group actions with the weak tracial Rokhlin property. Moreover, we discuss the stability of the property that all traces are quasidiagonal under taking crossed products of finite group actions with finite…
We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (1) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (2) Simple unital AH algebras with slow dimension growth and real rank zero. (3) C*-algebras with real rank zero or stable rank one. (4) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. Along the way, we give a systematic treatment of the derivation of direct limit decomposition…
In this paper we will introduce the tracial Rokhlin property for an inclusion of separable simple unital C*-algebras $P \subset A$ with finite index in the sense of Watatani, and prove theorems of the following type. Suppose that $A$ belongs to a class of C*-algebras characterized by some structural property, such as tracial rank zero in the sense of Lin. Then $P$ belongs to the same class. The classes we consider include:(1) Simple C*-algebras with real rank zero or stable rank one, (2) Simple C*-algebras with tracial rank zero or tracial rank less than or equal to one, (3) Simple C*-algebras…
We show that if $A$ is a simple (not necessarily unital) tracially $\mathcal{Z}$-absorbing C*-algebra and $α\colon G \to \mathrm{Aut} (A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then the crossed product $C^*(G, A,α)$ and the fixed point algebra $A^α$ are simple and tracially $\mathcal{Z}$-absorbing, and they are $\mathcal{Z}$-stable if, in addition, $A$ is separable and nuclear. The same conclusion holds for all intermediate C*-algebras of the inclusions $A^α\subseteq A$ and $A \subseteq C^*(G, A,α)$. We prove that if $A$ is a simple tracially $\mathc…
In this paper we define a Rokhlin property for automorphisms of non-unital C*-algebras and for endomorphisms. We show that the crossed product of a C*-algebra by a Rokhlin automorphism preserves absorption of a strongly self-absorbing C*-algebra, and use this result to deduce that the same result holds for crossed products by endomorphisms in the sense of Stacey. This generalizes earlier results of the second named author and W. Winter.
Let $A$ be a unital separable simple amenable $C^*$-algebra with finite tracial rank which satisfies the Universal Coefficient Theorem (UCT). Suppose $\af$ and $\bt$ are two automorphisms with the Rokhlin property that {induce the same action on the $K$-theoretical data of $A$.} We show that $\af$ and $\bt$ are strongly cocycle conjugate and uniformly approximately conjugate, that is, there exists a sequence of unitaries $\{u_n\}\subset A$ and a sequence of strongly asymptotically inner automorphisms $σ_n$ such that $$ \af={\rm Ad}\, u_n\circ σ_n\circ \bt\circ σ_n^{-1}\andeqn \lim_{n\to\infty}…
We study a tracial notion of Z-absorption for simple, unital C*-algebras. We show that if A is a C*-algebra for which this property holds then A has almost unperforated Cuntz semigroup, and if in addition A is nuclear and separable we show this property is equivalent to having A being Z-absorbing. We furthermore show that this property is preserved under forming certain crossed products by actions satisfying a tracial Rokhlin type property.
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 present a systematic study of the structure of crossed products and fixed point algebras by compact group actions with the Rokhlin property on not necessarily unital C*-algebras. Our main technical result is the existence of an approximate homomorphism from the algebra to its subalgebra of fixed points, which is a left inverse for the canonical inclusion. Upon combining this with results regarding local approximations, we show that a number of classes characterized by inductive limit decompositions with weakly semiprojective building blocks, are closed under formation of crossed products by…
Let $G$ be a finite group, $A$ a unital separable finite simple nuclear C*-algebra, and $α$ an action of $G$ on $A$. Assume that $A$ absorbs the Jiang-Su algebra $\mathcal{Z}$, the extremal boundary of the trace space of $A$ is compact and finite dimensional and that $α$ fixes any tracial state of $A$. Then tsr$(A \rtimes_αG) = 1$. In particular, when $A$ has a unique tracial state, we conclude it without above conditons on a tracial state space of $A$.
We show that formation of crossed products and passage to fixed point algebras by compact group actions with finite Rokhlin dimension preserve the following regularity properties: finite decomposition rank, finite nuclear dimension, and tensorial absorption of the Jiang-Su algebra, the latter in the formulation with commuting towers.
We introduce a general class of automorphisms of rotation algebras, the noncommutative Furstenberg transformations. We prove that fully irrational noncommutative Furstenberg transformations have the tracial Rokhlin property, which is a strong form of outerness. We conclude that crossed products by these automorphisms have stable rank one, real rank zero, and order on projections determined by traces (Blackadar's Second Fundamental Comparability Question). We also prove that several classes of simple quotients of the C*-algebras of discrete subgroups of five dimensional nilpotent Lie groups, co…
Let X be an infinite compact metric space with finite covering dimension and let h be a minimal homeomorphism of X. Let A be the associated crossed product C*-algebra. We show that A has tracial rank zero whenever the image of K_0 (A) in the affine functions on the tracial state space of A is dense. As a consequence, we show that these crossed product C*-algebras are in fact simple AH algebras with real rank zero. When X is connected and h is further assumed to be uniquely ergodic, then the above happens if and only if the rotation number associated to h has irrational values. By applying the …
University of Detroit Mercy（底特律 Mercy 大学）；Okanagan College（奥卡纳根学院）；East China Normal University（华东师范大学）；University of Oregon（俄勒冈大学）；University of Oxford（牛津大学）
We consider the tracial crossed product algebra $M=A\rtimesΛ$ arising from a trace preserving action $σ:Λ\curvearrowright A$ of a discrete group $Λ$ on a tracial von Neumann algebra $A$. For a unitary subgroup $\mathcal{G}\subset \mathcal{U}(M)$, we study when this $\mathcal{G}$ can be conjugated into $\mathcal{U}(A)\cdotΛ$ in $M$. We provide a general sufficient condition for this to happen. Our result generalizes a result of Ioana, Popa and Vaes, which treats the case when $M$ is the group von Neumann algebra $L(Λ)$.
Let A be a unital separable C*-algebra, and D a K_1-injective strongly self-absorbing C*-algebra. We show that if A is D-absorbing, then the crossed product of A by a compact second countable group or by Z or by R is D-absorbing as well, assuming the action satisfying a Rokhlin property. In the case of a compact Rokhlin action we prove a similar statement about approximate divisibility.
We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra, and a bijective correspondence between tracial states on the reduced crossed product and invariant tracial states on the underlying C*-algebra. In particular, the reduced crossed product is simple if and only if the underlying C*-algebra has no proper non-trivial invariant ideals, and the reduced crossed product has a unique tracial state if and on…
We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C*-algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C*-algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative C*-algeb…
For any countable discrete group $G$ with a reduced abelian subgroup of finite index, we construct an action $α$ of $G$ on the universal UHF algebra $\Qq$ using an infinite tensor product of permutation representations of $G$ and show that these actions possess some sort of Rokhlin property. The crossed product $\qag$ is then deduced to be tracially AF with a unique tracial state. We also compute the Elliott invariants in the case that $G$ is abelian.
We describe some of the forms of freeness of group actions on noncommutative C*-algebras that have been used, with emphasis on actions of finite groups. We give some indications of their strengths, weaknesses, applications, and relationships to each other. The properties discussed include the Rokhlin property, K-theoretic freeness, the tracial Rokhlin property, pointwise outerness, saturation, hereditary saturation, and the requirement that the strong Connes spectrum be the entire dual.
Let $X$ be a Cantor set, and let $A$ be a unital separable simple amenable $C$*-algebra with tracial rank zero which satisfies the Universal Coefficient Theorem, we use $C(X,A)$ to denote the set of all continuous functions from $X$ to $A$, let $α$ be an automorphism on $C(X,A)$. Suppose that $C(X,A)$ is $α$-simple and $[α]=[\mbox{id}_{1\otimes A}]$ in $KL(1\otimes A,1\otimes A)$, we show that $C(X,A)\rtimes_α\mathbb{Z}$ has tracial rank zero.
We give complete descriptions of the tracial states on both the universal and reduced crossed products of a C*-dynamical system consisting of a unital C*-algebra and a discrete group. In particular, we also answer the question of when the tracial states are in canonical bijection with the invariant tracial states on the original C*-algebra. This generalizes the unique trace property for discrete groups. The analysis simplifies greatly in various cases, for example when the conjugacy classes of the original group are all finite, and in other cases gives previously known results, for example whe…
