scholarly journals Graph algebras and orbit equivalence

2015 ◽  
Vol 37 (2) ◽  
pp. 389-417 ◽  
Author(s):  
NATHAN BROWNLOWE ◽  
TOKE MEIER CARLSEN ◽  
MICHAEL F. WHITTAKER
Keyword(s):  
Directed Graphs ◽  
Orbit Equivalence ◽  
Graph Algebras ◽  
Orbit Equivalent ◽  
Markov Shifts ◽  
Graph Algebra ◽  
Arbitrary Graphs ◽  
Reverse Implication

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their $C^{\ast }$-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs $E$ we construct a groupoid ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ from the graph algebra $C^{\ast }(E)$ and its diagonal subalgebra ${\mathcal{D}}(E)$ which generalises Renault’s Weyl groupoid construction applied to $(C^{\ast }(E),{\mathcal{D}}(E))$. We show that ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ recovers the graph groupoid ${\mathcal{G}}_{E}$ without the assumption that every cycle in $E$ has an exit, which is required to apply Renault’s results to $(C^{\ast }(E),{\mathcal{D}}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.

A dynamical characterization of diagonal-preserving -isomorphisms of graph -algebras

2017 ◽  
Vol 38 (7) ◽  
pp. 2401-2421 ◽  
Author(s):  
SARA E. ARKLINT ◽  
SØREN EILERS ◽  
EFREN RUIZ
Keyword(s):  
Directed Graphs ◽  
Sufficient Condition ◽  
Orbit Equivalence ◽  
Necessary And Sufficient Condition ◽  
Graph Algebras ◽  
Boundary Path ◽  
Necessary And Sufficient

We characterize when there exists a diagonal-preserving $\ast$-isomorphism between two graph $C^{\ast }$-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of ‘orbit equivalence’ between the boundary path spaces of the directed graphs $E$ and $F$ and show that this is a necessary and sufficient condition for the existence of a diagonal-preserving $\ast$-isomorphism between the graph $C^{\ast }$-algebras $C^{\ast }(E)$ and $C^{\ast }(F)$.

scholarly journals Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions

2015 ◽  
Vol 36 (5) ◽  
pp. 1557-1581 ◽  
Author(s):  
KENGO MATSUMOTO ◽  
HIROKI MATUI
Keyword(s):  
Zeta Functions ◽  
Periodic Points ◽  
Orbit Equivalence ◽  
Cohomology Groups ◽  
Borel Measures ◽  
Orbit Equivalent ◽  
Positive Elements ◽  
Markov Shifts ◽  
Sigma B

For continuously orbit equivalent one-sided topological Markov shifts $(X_{A},{\it\sigma}_{A})$ and $(X_{B},{\it\sigma}_{B})$, their eventually periodic points and cocycle functions are studied. As a result, we directly construct an isomorphism between their ordered cohomology groups $(\bar{H}^{A},\bar{H}_{+}^{A})$ and $(\bar{H}^{B},\bar{H}_{+}^{B})$. We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of their ordered cohomology groups, so that the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invariant under continuous orbit equivalence of one-sided topological Markov shifts.

IDENTITIES IN BIREGULAR LEFTMOST GRAPH VARIETIES OF TYPE (2,0)

2009 ◽  
Vol 02 (01) ◽  
pp. 1-17
Author(s):  
Apinant Anantpiniwatna ◽  
Tiang Poomsa-Ard
Keyword(s):  
Universal Algebra ◽  
Directed Graphs ◽  
Graph Algebras ◽  
Universal Algebras ◽  
Graph Algebra ◽  
Basic Concepts ◽  
Multiple Edges ◽  
Graph Varieties

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = ModgΣ where Σ is a subset of T(X) × T(X). A graph variety V' = ModgΣ' is called a biregular leftmost graph variety if Σ' is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety V if G satisfies s ≈ t for all G ∈ V. In this paper we characterize identities in each biregular leftmost graph variety. For identities, varieties and other basic concepts of universal algebra see e.g. [1].

scholarly journals Orbit equivalence of graphs and isomorphism of graph groupoids

2018 ◽  
Vol 123 (2) ◽  
pp. 239-248 ◽  
Author(s):  
Toke Meier Carlsen ◽  
Marius Lie Winger
Keyword(s):  
Directed Graphs ◽  
Periodic Points ◽  
Path Space ◽  
Orbit Equivalence ◽  
Graph Groupoids ◽  
Orbit Equivalent ◽  
Boundary Path ◽  
Isolated Points

We show that the groupoids of two directed graphs are isomorphic if and only if the two graphs are orbit equivalent by an orbit equivalence that preserves isolated eventually periodic points. We also give a complete description of the (topological) isolated points of the boundary path space of a graph. As a result, we are able to show that the groupoids of two directed graphs with finitely many vertices and no sinks are isomorphic if and only if the two graphs are orbit equivalent, and that the groupoids of the stabilisations of two such graphs are isomorphic if and only if the stabilisations of the graphs are orbit equivalent.

scholarly journals A short note on Cuntz splice from a viewpoint of continuous orbit equivalence of topological Markov shifts

2018 ◽  
Vol 123 (1) ◽  
pp. 91-100
Author(s):  
Kengo Matsumoto
Keyword(s):  
Short Note ◽  
Markov Shift ◽  
Orbit Equivalence ◽  
Irreducible Matrix ◽  
Orbit Equivalent ◽  
Markov Shifts ◽  
Topological Markov Shift ◽  
The One

Let $A$ be an $N\times N$ irreducible matrix with entries in $\{0,1\}$. We present an easy way to find an $(N+3)\times (N+3)$ irreducible matrix $\bar {A}$ with entries in $\{0,1\}$ such that the associated Cuntz-Krieger algebras ${\mathcal {O}}_A$ and ${\mathcal {O}}_{\bar {A}}$ are isomorphic and $\det (1 -A) = - \det (1-\bar {A})$. As a consequence, we find that two Cuntz-Krieger algebras ${\mathcal {O}}_A$ and ${\mathcal {O}}_B$ are isomorphic if and only if the one-sided topological Markov shift $(X_A, \sigma _A)$ is continuously orbit equivalent to either $(X_B, \sigma _B)$ or $(X_{\bar {B}}, \sigma _{\bar {B}})$.

Graph variety generated by linear terms

2019 ◽  
Vol 12 (05) ◽  
pp. 1950074
Author(s):  
C. Manyuen ◽  
P. Jampachon ◽  
T. Poomsa-ard
Keyword(s):  
Directed Graphs ◽  
Linear Term ◽  
Graph Algebras ◽  
Universal Algebras ◽  
Type Formula ◽  
Graph Algebra ◽  
Multiple Edges ◽  
Graph Varieties

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type [Formula: see text]. We say that a graph [Formula: see text] satisfies a term equation [Formula: see text] if the corresponding graph algebra [Formula: see text] satisfies [Formula: see text]. The set of all term equations [Formula: see text], which the graph [Formula: see text] satisfies, is denoted by [Formula: see text]. The class of all graph algebras satisfy all term equations in [Formula: see text] is called the graph variety generated by [Formula: see text] denoted by [Formula: see text]. A term is called a linear term if each variable which occurs in the term, occurs only once. A term equation [Formula: see text] is called a linear term equation if [Formula: see text] and [Formula: see text] are linear terms. This paper is devoted to a thorough investigation of graph varieties defined by linear term equations. In particular, we give a complete description of rooted graphs generating a graph variety described by linear term equations.

scholarly journals Lebesgue orbit equivalence of multidimensional Borel flows: A picturebook of tilings

2016 ◽  
Vol 37 (6) ◽  
pp. 1966-1996
Author(s):  
KONSTANTIN SLUTSKY
Keyword(s):  
Lebesgue Measure ◽  
Probability Measures ◽  
Orbit Equivalence ◽  
Orbit Equivalent

The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue orbit equivalence, by which we mean an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non-smooth $\mathbb{R}^{d}$-flows are shown to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.

scholarly journals Syntactic semigroups and graph algebras

2000 ◽  
Vol 62 (3) ◽  
pp. 471-477 ◽  
Author(s):  
A. V. Kelarev ◽  
O. V. Sokratova
Keyword(s):  
Directed Graphs ◽  
Graph Algebras

We describe all directed graphs with graph algebras isomorphic to syntactic semigroups of languages.

scholarly journals Graph varieties containing Murskii's groupoid

1989 ◽  
Vol 39 (2) ◽  
pp. 265-276
Author(s):  
R. Pöschel
Keyword(s):  
Finitely Based ◽  
Graph Algebras ◽  
Undirected Graphs ◽  
Graph Algebra ◽  
Subvariety Lattice ◽  
As Graph ◽  
Graph Varieties ◽  
Nonfinitely Based

In this paper varieties are investigated which are generated by graph algebras of undirected graphs and—in most cases—contain Murskii's groupoid (that is the graph algebra of the graph with two adjacent vertices and one loop). Though these varieties are inherently nonfinitely based, they can be finitely based as graph varieties (finitely graph based) like, for example, the varitey generated by Murskii's groupoid. Many examples of nonfinitely based graph varities containing Murskii's groupoid are given, too. Moreover, the coatoms in the subvariety lattice of the graph variety of all undirected graphs are described. There are two coatoms and they are finitely graph based.

scholarly journals A FUNCTORIAL APPROACH TO THE C*-ALGEBRAS OF A GRAPH

2002 ◽  
Vol 13 (03) ◽  
pp. 245-277 ◽  
Author(s):  
JACK SPIELBERG
Keyword(s):  
Structural Properties ◽  
Inductive Limit ◽  
Directed Graphs ◽  
Graph Algebras ◽  
C Algebras ◽  
Injective Homomorphisms ◽  
Functorial Approach

A functor from the category of directed trees with inclusions to the category of commutative C*-algebras with injective *-homomorphisms is constructed. This is used to define a functor from the category of directed graphs with inclusions to the category of C*-algebras with injective *-homomorphisms. The resulting C*-algebras are identified as Toeplitz graph algebras. Graph algebras are proved to have inductive limit decompositions over any family of subgraphs with union equal to the whole graph. The construction is used to prove various structural properties of graph algebras.

Sign in / Sign up

Export Citation Format

Share Document