Let $X$ be a compact metric space and let $\af$ be a homeomorphism on $X.$ Related to a theorem of Pimsner, we show that $C(X)\rtimes_{\af}\Z$ can be embedded into a unital simple AF-algebra if and only if there is a strictly positive $\af$-invariant Borel probability measure.
Suppose that $Λ$ is a $\Z^d$ action on $X.$ If $C(X)\rtimes_Λ\Z$ can be embedded into a unital simple AF-algebra, then there must exist a strictly positive $Λ$-invariant Borel probability measure. We show that, if in addition, there is a generator $\af_1$ of $Λ$ such that $(X, \af_1)$ is minimal and unique ergodic, then $C(X)\rtimes_Λ\Z^d$ can be embedded into a unital simple AF-algebra with a unique tracial state.
Let $A$ be a unital separable amenable simple \CA with tracial rank zero and with a unique tracial state which satisfies the Universal Coefficient Theorem and let $G$ be a finitely generated discrete abelian group. Suppose $Λ: G\to Aut(A)$ is a \hm. Then $A\rtimes_Λ G$ can always be embedded into a unital simple AF-algebra.
We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for α. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.
Let a discrete group $G$ act on a unital simple C$^*$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes_{α,r}H$ between subgroups of $G$ and C$^*$-algebras $B$ satisfying $A\subseteq B \subseteq A\rtimes_{α,r}G$, where $A\rtimes_{α,r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,α,σ)$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes^σ_{α,r}G$.
We provide a reference for basic categorial properties of the categories of (possibly non-unital) $\mathbb{C}$-linear $*$-categories or $C^{*}$-categories, and (not necessarily unit-preserving) functors. Generalizing the classical case of algebras with $G$-action, we extend the construction of crossed products to categories with $G$-action. We will show that the crossed product functor preserves exact sequences and excisive squares and sends weak equivalences to equivalences.
We prove that the crossed product A x G of a separable, unital, quasidiagonal C*- algebra A by a discrete, countable, amenable, maximally almost periodic group G is quasidiagonal, provided that the action is almost periodic.
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 …
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 al…
If $α$ is an amenable action of a discrete group $G$ on a unital C*-algebra $A$, then the crossed-product C*-algebra $A\rtimes_αG$ has the weak expectation property if and only if $A$ has this property.
It is shown that the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism is purely infinite when it is simple, provided the endomorphism does not leave a trace state invariant and maps the unit to a full projection.
Laca constructed a minimal automorphic dilation for every semigroup dynamical system arising from an action of an Ore semigroup by injective endomorphisms of a unital $C^*$-algebra. Here we show that the semigroup crossed product with its action by inner endomorphisms given by the implementing isometries has as minimal automorphic dilation the group crossed product of the original dilation. Applications include recent examples studied by Cuntz and the second named author.
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 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…
We prove that the crossed product AxG of a unital finitely generated MF algebra A by a discrete finitely generated amenable residually finite group G is an MF algebra, provided that the action is almost periodic. This generalizes a result of Hadwin and Shen. We also construct two examples of crossed product C*-algebras whose BDF Ext semigroups are not groups.
Building on an argument by Toms and Winter, we show that if $A$ is a simple, separable, unital, $\mathcal{Z}$-stable C*-algebra, then the crossed product of $C(X,A)$ by an automorphism is also Z-stable, provided that the automorphism induces a minimal homeomorphism on $X$. As a consequence, we observe that if $A$ is nuclear and purely infinite then the crossed product is a Kirchberg algebra.
Let $X$ be a compact metric space and let $Λ$ be a $\Z^k$ ($k\ge 1$) action on $X.$ We give a solution to a version of Voiculescu's problem of AF-embedding: The crossed product $C(X)\rtimes_Λ\Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $Λ$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $Λ: G\to Aut(C)$ be a monomorphism. We show that $C\rtimes_Λ G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $Λ$-invariant tracial state.
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 study uniform perturbations of crossed product C$^*$-algebras by amenable groups. Given a unital inclusion of C$^*$-algebras $C\subseteq D$ and sufficiently close separable intermediate C$^*$-subalgebras $A$, $B$ for this inclusion with a conditional expectation from $D$ onto $B$, if $A=C\rtimes G$ with $G$ discrete amenable, then $A$ and $B$ are isomorphic. Furthermore, if $C\subseteq D$ is irreducible, then $A=B$.
We consider a certain class of unital simple stably finite C^*-algebras which absorb the Jiang-Su algebra Z tensorially. Under a mild assumption, we show that the crossed product of a C^*-algebra in this class by a strongly outer action of Z^N or a finite group is Z-stable. As an application, we also prove that all strongly outer actions of Z^2 on Z are mutually cocycle conjugate.
Let $A$ be a unital AH-algebra and let $α\in Aut(A)$ be an automorphism. A necessary condition for $A\rtimes_α\Z$ being embedded into a unital simple AF-algebra is the existence of a faithful tracial state. If in addition, there is an automorphism $κ$ with $κ_{*1}=-{\rm id}_{K_1(A)}$ such that $α\circ κ$ and $κ\circ \af$ are asymptotically unitarily equivalent, then $A\rtimes_{\af}\Z$ can be embedded into a unital simple AF-algebra. Consequently, in the case that $A$ is a unital AH-algebra (not necessarily simple) with torsion $K_1(A),$ $A\rtimes_α\Z$ can be embedded into a unital simple AF-al…
C. Schochet shows Künneth theorem for the $C^*$-algebras in the smallest class of nuclear $C^*$-algebras which contains the separable Type I algebras and is closed under some operations. We calculate the $K$-theory for the crossed product of the infinite tensor product of a unital $C^*$-algebra in this class by the shift. In particular, we calculate the $K$-theory of the lamplighter group $C^*$-algebra.
Following Jansen and Waldmann, and Kajiwara and Watatani, we shall introduce notions of coactions of a finite dimensional $C^*$-Hopf algebra on a Hilbert $C^*$-bimodule of finite type in the sense of Kajiwara and Watatani and define their crossed product. We shall investigate their basic properties and show that the strong Morita equivalence for coactions preserves the Rohlin property for coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras.
We define centrally large subalgebras of simple unital C*-algebras, strengthening the definition of large subalgebras in previous work. We prove that if A is any infinite dimensional simple separable unital C*-algebra which contains a centrally large subalgebra with stable rank one, then A has stable rank one. We also prove that large subalgebras of crossed product type are automatically centrally large. We use these results to prove that if X is a compact metric space which has a surjective continuous map to the Cantor set, and h is a minimal homeomorphism of X, then C* (Z, X, h) has stable r…
We generalize the main result of Kamalov and show that if $G$ is an amenable discrete group with an action $α$ on a finite nuclear unital $C^*$-algebra $A$ such that the reduced crossed product $A\rtimes_{α,r} G$ has property $T$, then $G$ is finite and $A$ is finite dimensional. As an application, an infinite discrete group $H$ is non-amenable if and only if the uniform Roe algebra $C^*_u(H)$ has property $T$.
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 $Γ$, 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 $ϕ$ from a unital $C^*$-algebra $A$ to the twisted join of $A$ and $C^*(Γ)$ 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 …
We describe the structure of the irreducible representations of crossed products of unital C*-algebras by actions of finite groups in terms of irreducible representations of the C*-algebras on which the groups act. We then apply this description to derive a characterization of irreducible representations of crossed-products by finite cyclic groups in terms of representations of the C*-algebra and its fixed point subalgebra. These results are applied to crossed-products by the permutation group on three elements and illustrated by various examples.
We describe the representation theory of C*-crossed-products of a unital C*-algebra A by the cyclic group of order 2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of A. We characterize each class in term of the restriction of the representations to the fixed point C*-subalgebra of A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators.
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.
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…
Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let α\colon G \to Aut (A) be an action of G on A which has the weak tracial Rokhlin property. Let A^α be the fixed point algebra. Then the radius of comparison satisfies rc (A^α) \leq rc (A) and rc ( C* (G, A, α) ) \leq ( 1 / card (G) ) rc (A). The inclusion of A^α in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu (A^α) to the fixed points of the purely positive part of Cu (A), and the purely positive part of Cu ( C* (G, A, α) ) is isomorphic to this semigroup…
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 $A$.
Let $G$ be a discrete group acting on a unital $C^*$-algebra $\mathcal{A}$ by $*$-automorphisms. We characterize (in terms of the dynamics) when the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_r G$ has a unique conditional expectation, and when it has a unique pseudo-expectation (in the sense of Pitts). Likewise for the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes G$. As an application, we (slightly) strengthen results of Kishimoto and Archbold-Spielberg concerning $C^*$-simplicity of $\mathcal{A} \rtimes_r G$.
Let $P \subset A$ be an inclusion of separable unital C*-algebras with finite Watatani index. Suppose that $E \colon A \rightarrow P$ has the Rokhlin property, that is, there is a projection $e \in A' \cap A^\infty$ such that $E^\infty(e) = ({\rm Index}E)^{-1}1$. We show that if $A$ has nuclear dimension $n$, then $P$ has nuclear dimension less than or equal to $n$. In particular, if an action $α$ of a finite group $G$ on $A$ has the Rokhlin property, then the nuclear dimension of the crossed product algebra $A \rtimes_αG$ is less than or equal to that of $A$.
Starting from an arbitrary endomorphism δof a unital C*-algebra A we construct a crossed product. It is shown that the natural construction depends not only on the C*-dynamical system (A,δ) but also on the choice of an ideal J orthogonal to Ker δ.
The notions of Busby-Smith and Green type twisted actions are extended to discrete unital inverse semigroups. The connection between the two types, and the connection with twisted partial actions, are investigated. Decomposition theorems for the twisted crossed products are given.
For a twisted partial action Θof a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A X_ΘG is proved to be associative. Given a G-graded k-algebra B = \oplus_{g\in G}\B_g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B_1 X_ΘG for some twisted partial action of G on B_1. The equality B_g\B_{g^{-1}}B_g = \B_g for all g\in G is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is …
We consider two twisted actions of a countable discrete group on $σ$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions is irreducible, that is, the relative commutant of one of the $σ$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.
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.
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.
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 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…
Let $G$ be a discrete group acting on a unital $C^*$-algebra $\mathcal{A}$ by $*$-automorphisms. In this note, we show that the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_r G$ has the pure extension property (so that every pure state on $\mathcal{A}$ extends uniquely to a pure state on $\mathcal{A} \rtimes_r G$) if and only if $G$ acts freely on $\mathcal{\widehat{A}}$, the spectrum of $\mathcal{A}$. The same characterization holds for the inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes G$. This generalizes what was already known for $\mathcal{A}$ abelian.
Let $1 \in A \subset B$ be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute topological stable rank of $B$ ($= \tsr(B)$) when $A$ has topological stable rank one. We show that $\tsr(B) \leq 2$ when $A$ is a tsr boundedly divisible algebra, in particular, $A$ is a C*-minimal tensor product $UHF \otimes D$ with $\tsr(D) = 1$. When $G$ is a finite group and $α$ is an action of $G$ on UHF, we know that a crossed product algebra $UHF \rtimes_αG$ has topological stable rank less than or equal to two. These results are affirmative datum to a generalization of a ques…
We introduce the labelling map and the quasi-free action of a locally compact abelian group on a graph $C^*$-algebra of a row-finite directed graph. Some necessary conditions for embedding the crossed product to an $AF$ algebra are discussed, and one sufficient condition is proved that if the row-finite directed graph is constructed by possibly attaching some 1-loops to a row-finite directed graph whose each weak connected component is a rooted (possibly infinite) directed tree, and the labelling map is almost proper, which is proved to be a reasonable generalization of the earlier case, then …
Starting from an arbitrary endomorphism αof a unital C*-algebra A we construct a crossed product. It is shown that the natural construction depends not only on the C*-dynamical system (A,α) but also on the choice of an ideal orthogonal to kernel of α. The article gives an explicit description of the internal structure of this crossed product and, in particular, discusses the interrelation between relative Cuntz-Pimsner algebras and partial isometric crossed products. We present a canonical procedure that reduces any given C*-correspondence to the 'smallest' C*-correspondence yielding the same …
We show that for a closed embedding $\mathbb{H}\le \mathbb{G}$ of locally compact quantum groups (LCQGs) with $\mathbb{G}/\mathbb{H}$ admitting an invariant probability measure, a unitary $\mathbb{G}$-representation is type-I if its restriction to $\mathbb{H}$ is. On a related note, we also prove that if an action $\mathbb{G}\circlearrowright A$ of an LCQG on a unital $C^*$-algebra admits an invariant state then the full group algebra of $\mathbb{G}$ embeds into the resulting full crossed product (and into the multiplier algebra of that crossed product if the original algebra is not unital). W…
We prove that the infinite tensor power of a unital separable C*-algebra absorbs the Jiang-Su algebra Z tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no unital separable C*-algebras without characters known to lie outside it. This category moreover contains the vast majority of our stock-in-trade separable amenable C*-algebras, and is closed under passage to separable superalgebras and quotients, and hence to unital tensor products, unital direct limits, and cr…
We prove that if A is a σ-unital exact C*-algebra of real rank zero, then every state on K_0(A) is induced by a 2-quasitrace on A. This yields a generalisation of Rainone's work on pure infiniteness and stable finiteness of crossed products to the non-unital case. It also applies to k-graph algebras associated to row-finite k-graphs with no sources. We show that for any k-graph whose C*-algebra is unital and simple, either every twisted C*-algebra associated to that k-graph is stably finite, or every twisted C*-algebra associated to that k-graph is purely infinite. Finally we provide sufficien…
Let $G$ be an inductive limit of finite cyclic groups and let $A$ be a unital simple projectionless C*-algebra with $K_1(A) \cong G$ and with a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in $\Aut(A)/\WInn(A)$ are conjugate, where $\WInn(A)$ means the subgroup of $\Aut(A)$ consisting of automorphisms which are inner in the tracial representation. In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple AT-algeb…
