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.
There are many different crossed products by an endomorphism of a C*-algebra, and constructions by Exel and Stacey have proved particularly useful. Here we show that every Exel crossed product is isomorphic to a Stacey crossed product, though by a different endomorphism of a different C*-algebra. We apply this result to a variety of Exel systems, including those associated to shifts on the path spaces of directed graphs.
There are many different crossed products by an endomorphism of a C*-algebra, and constructions by Exel and Stacey have proved particularly useful. Here we show that every Exel crossed product is isomorphic to a Stacey crossed product (though by a different endomorphism of a different C*-algebra), that every Stacey crossed product is an Exel crossed product, and answer some questions raised by Ionescu and Muhly. While this manuscript is not yet in its final form(s) several people have already used our results, so we are posting it now.
We consider Exel's new construction of a crossed product of a C*-algebra A by an endomorphism α. We prove that this crossed product is universal for an appropriate family of covariant representations, and we show that it can be realised as a relative Cuntz-Pimsner algbera. We describe a necessary and sufficient condition for the canonical map from A into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product.
Larsen has recently extended Exel's construction of crossed products from single endomorphisms to abelian semigroups of endomorphisms, and here we study two families of her crossed products. First, we look at the natural action of the multiplicative semigroup $\mathbb{N}^\times$ on a compact abelian group $Γ$, and the induced action on $C(Γ)$. We prove a uniqueness theorem for the crossed product, and we find a class of connected compact abelian groups $Γ$ for which the crossed product is purely infinite simple. Second, we consider some natural actions of the additive semigroup $\mathbb{N}^2$ …
We use the boundary-path space of a finitely-aligned k-graph Λto construct a compactly-aligned product system X, and we show that the graph algebra C^*(Λ) is isomorphic to the Cuntz-Nica-Pimsner algebra NO(X). In this setting, we introduce the notion of a crossed product by a semigroup of partial endomorphisms and partially-defined transfer operators by defining it to be NO(X). We then compare this crossed product with other definitions in the literature.
We introduce crossed products of a $C^*$-algebra $A$ by a completely positive map $\varrho:A\to A$ relative to an ideal in $A$. They generalize various crossed products by endomorphisms when $\varrho$ is multiplicative. When $A$ is commutative they include $C^*$-algebras associated to Markov operators by Ionescu, Muhly, Vega, and to topological relations by Brenken, but in general they are not modeled by topological quivers popularized by Muhly and Tomforde. We show that Exel's crossed product $A\rtimes_{α,L} \mathbb{N}$, generalized to the case where $A$ is not necessarily unital, is the cros…
An integer matrix $A\in M_d(\Z)$ induces a covering $σ_A$ of $\T^d$ and an endomorphism $α_A:f\mapsto f\circ σ_A$ of $C(\T^d)$ for which there is a natural transfer operator $L$. In this paper, we compute the KMS states on the Exel crossed product $C(\T^d)\rtimes_{α_A,L}\N$ and its Toeplitz extension. We find that $C(\T^d)\rtimes_{α_A,L}\N$ has a unique KMS state, which has inverse temperature $β=\log|\det A|$. Its Toeplitz extension, on the other hand, exhibits a phase transition at $β=\log|\det A|$, and for larger $β$ the simplex of KMS$_β$ states is isomorphic to the simplex of probability …
We found that if $u$ and $v$ are any two unitaries in a unital $C^*$-algebra with $\|uv-vu\|<2$ such that $uvu^*v^*$ commutes with $u$ and $v,$ then the \SCA\, $A_{u,v}$ generated by $u$ and $v$ is isomorphic to a quotient of the rotation algebra $A_θ$ provided that $A_{u,v}$ has a unique tracial state. We also found that the Exel trace formula holds in any unital $C^*$-algebra. Let $θ\in (-1/2, 1/2)$ be a rational number. We prove the following: For any $\ep>0,$ there exists $\dt>0$ satisfying the following: if $u$ and $v$ are two unitary matrices such that $$ \|uv-e^{2πiθ}vu\|<\dt\andeqn {1\…
For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product $C^*$-algebras $\cros$ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical imbedding of $C(X)$ i…
Partial actions of discrete groups on $C^*$-algebras and the associated crossed products have been studied by Exel and McClanahan. We characterise these crossed products in terms of the spectral subspaces of the dual coaction, generalising and simplifying a theorem of Exel for single partial automorphisms. We then use this characterisation to identify the Cuntz algebras and the Toeplitz algebras of Nica as crossed products by partial actions.
The paper presents a construction of the crossed product of a C*-algebra by a commutative semigroup of bounded positive linear maps generated by partial isometries. In particular, it generalizes Antonevich, Bakhtin, Lebedev's crossed product by an endomorphism, and is related to Exel's interactions. One of the main goals is the Isomorphism Theorem established in the case of actions by endomorphisms.
We propose a generalisation of Exel's crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler's theory of Toeplitz and Cuntz-Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz-Pimsner algebras of product systems.
Given a dynamical system $(A,\al)$ where $A$ is a unital $\ca$-algebra and $\al$ is a (possibly non-unital) *-endomorphism of $A$, we examine families $(π,\{T_i\})$ such that $π$ is a representation of $A$, $\{T_i\}$ is a Toeplitz-Cuntz family and a covariance relation holds. We compute a variety of non-selfadjoint operator algebras that depend on the choice of the covariance relation, along with the smallest $\ca$-algebra they generate, namely the $\ca$-envelope. We then relate each occurrence of the $\ca$-envelope to (a full corner of) an appropriate twisted crossed product. We provide a cou…
We introduce an algebraic version of the Katsura $C^*$-algebra of a pair $A,B$ of integer matrices and an algebraic version of the Exel-Pardo $C^*$-algebra of a self-similar action on a graph. We prove a Graded Uniqueness Theorem for such algebras and construct a homomorphism of the latter into a Steinberg algebra that, under mild conditions, is an isomorphism. Working with Steinberg algebras over non-Hausdorff groupoids we prove that in the unital case, our algebraic version of Katsura $C^*$-algebras are all isomorphic to Steinberg algebras.
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 $A\rtimes^σ_{α,r}G$.
In this paper we extend the constructions of Boava and Exel to present the C*-algebra associated with an injective endomorphism of a group with finite cokernel as a partial group algebra and consequently as a partial crossed product. With this representation we present another way to study such C*-algebras, only using tools from partial crossed products.
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 …
We prove a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type. To this end, we introduce Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product system of 'regular' C*-correspondences. Under a certain aperiodicity condition on the latter, we obtain the uniqueness theorem and a simplicity criterion for the algebras in question. These results generalize the corresponding ones for crossed products by discrete groups, due to Archbold and Spielberg, and for Exel's crossed products, due to Exel and Vershik…
In this work we present a new definition to the Partial Crossed Product by actions of inverse semigroups in a C^*-algebra, without using the covariant representations as Sieben did in [5]. Also we present an isomorphism between the partial crossed products by partial actions of a group G and the partial crossed product by actions of S(G), an inverse semigroup associated to G introduced by Exel in [2].
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 study simplicity and pure infiniteness criteria for C*-algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over etale not necessarily Hausdorff groupoids. Inspired by recent work of Exel and Pitts, we introduce essential crossed products for which there are such criteria. In our approach the major role is played by a generalised expectation with values in the local multiplier algebra. We give a long list of equivalent conditions characterising when the essential and reduced C*-algebras coincide. Our most general simplicity and pure infiniteness criteria…
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.
We study the Toeplitz algebra $\TT(\N\rtimes\N^\times)$ and three quotients of this algebra: the $C^*$-algebra $\qn$ recntly introduced by Cuntz, and two new ones, which we call the additive and multiplicative boundary quotients. These quotients are universal for Nica-covariant representations of $\N\rtimes\N^\times$ satisfying extra relations, and can be realised as partial crossed products. We use the structure theory for partial crossed products to prove a uniqueness theorem for the additive boundary quotient, and use the recent analysis of KMS states on $\TT(\nxnx)$ to describe the KMS sta…
Using the strong relation between coactions of a discrete group G on C*-algebras and Fell bundles over G, we prove a new version of Mansfield's imprimitivity theorem for coactions of discrete groups. Our imprimitivity theorem works for the universally defined full crossed products and arbitrary subgroups of G, as opposed to the usual theory which uses the spatially defined reduced crossed products and normal subgroups of G. Moreover, our theorem factors through the usual one by passing to appropriate quotients. As applications we show that a Fell bundle over a discrete group is amenable in the…
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.
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.
In this article, we use Exel's construction to associate a C*-algebra to every shift space. We show that it has the C*-algebra defined in [Carlsen and Matsumoto: Some remarks on the C*-algebras associated with subshifts] as a quotient, and possesses properties indicating that it can be thought of as the universal C*-algebra associated to a shift space. We also consider its representations, relationship to other C*-algebras associated to shift spaces, show that it can be viewed as a generalization of the universal Cuntz-Krieger algebra, discuss uniqueness and a faithful representation, provide …
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.
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.
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 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…
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…
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…
