We show that certain C*-algebras which have been studied among others by Arzumanian, Vershik, Deaconu, and Renault in connection to a measure preserving transformation of a measure space and/or to a covering map of a compact space are special cases of the endomorphism crossed-product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general Theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity.
Extending the work of Cuntz and Vershik, we develop a general notion of independence for commuting group endomorphisms. Based on this concept, we initiate the study of irreversible algebraic dynamical systems, which can be thought of as irreversible analogues of the dynamical systems considered by Schmidt. To each irreversible algebraic dynamical system, we associate a universal C*-algebra and show that it is a UCT Kirchberg algebra under natural assumptions. Moreover, we discuss the structure of the core subalgebra, which is closely related to generalised Bunce-Deddens algebras in the sense o…
Since their inception in the 30's by von Neumann, operator algebras have been used in shedding light in many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two was sought since their emergence in the late 60's. We connect these seemingly separate type of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^*$-algebras with additional $C^*$-algebraic structure. Our approach naturally applies to algebras arising from $C^*$-correspondences …
Given a spectral triple on a $C^*$-algebra $\mathcal A$ together with a unital injective endomorphism $α$, the problem of defining a suitable crossed product $C^*$-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and of Hawkins, Skalski, White and Zacharias, and on our previous papers. The embedding of $α(\mathcal A)$ in $\mathcal A$ can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricit…
A countable family of $*$-commuting surjective, non-injective local homeomorphisms of a compact Hausdorff space $X$ gives rise to an action $θ$ of a countably generated, free abelian monoid $P$. For such a triple $(X,P,θ)$, which we call an irreversible $*$-commutative dynamical system, we construct a universal $C^*$-algebra $\mathcal{O}[X,P,θ]$. Within this setting we show that the following four conditions are equivalent: $(X,P,θ)$ is topologically free, $C(X) \subset \mathcal{O}[X,P,θ]$ has the ideal intersection property, the natural representation of $\mathcal{O}[X,P,θ]$ on $\ell^2(X)$ is…
A Fejér-type theorem is proved within the framework of $C^*$-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.
We introduce algebraic dynamical systems, which consist of an action of a right LCM semigroup by injective endomorphisms of a group. To each algebraic dynamical system we associate a C*-algebra and describe it as a semigroup C*-algebra. As part of our analysis of these C*-algebras we prove results for right LCM semigroups. More precisely we discuss functoriality of the full semigroup C*-algebra and compute its K-theory for a large class of semigroups. We introduce the notion of a Nica-Toeplitz algebra of a product system over a right LCM semigroup, and show that it provides a useful alternativ…
This paper is a survey of our recent work on operator algebras associated to dynamical systems that lead to classification results for the systems in terms of algebraic invariants of the operator algebras.
We associate to each discrete partial dynamical system a universal C*-algebra generated by partial isometries satisfying relations given by a Boolean algebra connected to the discrete partial dynamical system in question. We show that for symbolic dynamical systems like one-sided and two-sided shift spaces and topological Markov chains with an arbitrary state space the C*-algebras usually associated to them, can be obtained in this way. As a consequence of this, we will be able to show that for two-sided shift spaces having a certain property, the crossed product of the two-sided shift space i…
A $C^*$-symbolic dynamical system $({\cal A}, ρ, Σ)$ is a finite family $\{ρ_α\}_{α\inΣ}$ of endomorphisms of a $C^*$-algebra ${\cal A}$ with some conditions. It yields a $C^*$-algebra ${\cal O}_ρ$ from an associated Hilbert $C^*$-bimodule. In this paper, we will extend the notion of $C^*$-symbolic dynamical system to $C^*$-textile dynamical system $({\cal A}, ρ, η, {Σ^ρ}, {Σ^η}, κ)$ which consists of two $C^*$-symbolic dynamical systems $({\cal A}, ρ, {Σ^ρ})$ and $({\cal A}, η, {Σ^η})$ with certain commutation relations $κ$ between their endomorphisms $\{ρ_α\}_{α\in Σ^ρ}$ and $\{η_a \}_{a \in…
The general class of the graded Lie algebras is defined. These algebras could be constructed using an arbitrary dynamical systems with discrete time and with invarinat measure. In this papers we consider the case of the central extension of Lie algebras which corresponds to the ordinary crossed product (as associative algebra) - series A. The structure of those Lie algebras is similar to Kac-Moody algebras, and these are a special case of so called algebras with continuous root system which were introduced by author with M.Saveliev in 90-th. The central extension open a new possibilty in algeb…
In this paper, we realize C*-algebras of generalized Boolean dynamical systems as partial crossed products. Reciprocally, we give some sufficient conditions for a partial crossed product to be isomorphic to a C*-algebra of a generalized Boolean dynamical system. As an application, we show that gauge-invariant ideals of C*-algebras of generalized Boolean dynamical systems are themselves C*-algebras of generalized Boolean dynamical system.
We have introduced a notion of $C^*$-symbolic dynamical system in [K. Matsumoto: Actions of symbolic dynamical systems on $C^*$-algebras, to appear in J. Reine Angew. Math.], that is a finite family of endomorphisms of a $C^*$-algebra with some conditions. The endomorphisms are indexed by symbols and yield both a subshift and a $C^*$-algebra of a Hilbert $C^*$-bimodule. The associated $C^*$-algebra with the $C^*$-symbolic dynamical system is regarded as a crossed product by the subshift. We will study a simplicity condition of the $C^*$-algebras of the $C^*$-symbolic dynamical systems. Some ex…
In this paper, we show that the set of all ideals of the C*-algebras of a singly generated dynamical system corresponds bijectively to the set of all subsets of the product of the space of the system and the circle satisfying three conditions.
Pure infiniteness (in sense of E.Kirchberg and M.Rørdam) is considered for C*-algebras arising from singly generated dynamical systems. In particular, Cuntz-Krieger algebras and their generalizations, i.e., graph-algebras and O_A of an infinite matrix A, admit characterizations of pure infiniteness. As a consequence, these generalized Cuntz-Krieger algebras are traceless if and only if they are purely infinite. Also, a characterization of AF-algebras among these C*-algebras is given. In the case of graph-algebras of locally finite graphs, characterizations of stability are obtained.
We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis relating to work of Arveson from the sixties, and extends related work of Kakariadis and Katsoulis.
In this survey, we study the relations between amenability (resp. amenability at infinity) of C*-dynamical systems and equality or nuclearity (resp. exactness) of the corresponding crossed products.
We use a characterization of the graph C*-algebras C*(E) as the Stacey crossed product C*(E)^γ\times_{β_E} N, to study its ideal properties in terms of the (non-classical) C*-dynamical system (C*(E)^γ, β_E)
The goal of these notes is to present the C*-algebra $C^*(B,L,θ)$ of a Boolean dynamical system $(B,L,θ)$, that generalizes the $C^*$-algebra associated to Labelled graphs introduced by Bates and Pask, and to determine its simplicity, its gauge invariant ideals, as well as compute its K-Theory
Connections between partial dynamcial systems, a generalized notion of partial dynamical systems defined by nested sequences of partial homeomorphisms, and triangular AF algebras which admit an integer-valued cocycle are established.
We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain subalgebra to discuss a Shubin trace formula.
Evans-Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in \cite{Ma}. It is shown that these flows are unital and covariant. Ergodicity of the flows for the semigroups associated with partial states is also discussed.
