scholarly journals On Embedding of the Bratteli Diagram into a Surface

2007 ◽  
pp. 211-227
Author(s):  
Igor V. Nikolaev
Keyword(s):  
Bratteli Diagram

Bratteli–Vershik models for partial actions of ℤ

2017 ◽  
Vol 28 (10) ◽  
pp. 1750073 ◽  
Author(s):  
Thierry Giordano ◽  
Daniel Gonçalves ◽  
Charles Starling
Keyword(s):  
Cantor Set ◽  
Bratteli Diagram ◽  
Partial Action ◽  
Dense Orbits

Let [Formula: see text] and [Formula: see text] be open subsets of the Cantor set with nonempty disjoint complements, and let [Formula: see text] be a homeomorphism with dense orbits. Building on the ideas of Herman, Putnam and Skau, we show that the partial action induced by [Formula: see text] can be realized as the Vershik map on an ordered Bratteli diagram, and that any two such diagrams are equivalent.

On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type

2013 ◽  
Vol 56 (1) ◽  
pp. 136-147
Author(s):  
Radu-Bogdan Munteanu
Keyword(s):  
Equivalence Relation ◽  
Product Type ◽  
Bratteli Diagram ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Property A

AbstractProduct type equivalence relations are hyperfinitemeasured equivalence relations, which, up to orbit equivalence, are generated by product type odometer actions. We give a concrete example of a hyperfinite equivalence relation of non-product type, which is the tail equivalence on a Bratteli diagram. In order to show that the equivalence relation constructed is not of product type we will use a criterion called property A. This property, introduced by Krieger for non-singular transformations, is defined directly for hyperfinite equivalence relations in this paper.

scholarly journals Construction and pure infiniteness of $C^*$-algebras associated with lambda-graph systems

2005 ◽  
Vol 97 (1) ◽  
pp. 73 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Hilbert Space ◽  
Bratteli Diagram ◽  
C Algebra ◽  
C Algebras ◽  
Shift Transformation ◽  
Labeled Graphs ◽  
Graph System ◽  
Creation Operators

A $\lambda$-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In [16] the author has introduced a $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ associated with a $\lambda$-graph system $\mathfrak{L}$ by using groupoid method as a generalization of the Cuntz-Krieger algebras. In this paper, we concretely construct the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ by using both creation operators and projections on a sub Fock Hilbert space associated with $\mathfrak{L}$. We also introduce a new irreducible condition on $\mathfrak{L}$ under which the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ becomes simple and purely infinite.

Classification of AF Flows

2003 ◽  
Vol 46 (2) ◽  
pp. 164-177 ◽  
Author(s):  
Andrew J. Dean
Keyword(s):  
Dynamical Systems ◽  
Automorphism Group ◽  
Bratteli Diagram ◽  
Finite Dimensional

AbstractAn AF flow is a one-parameter automorphism group of an AF C*-algebra A such that there exists an increasing sequence of invariant finite dimensional sub-C*-algebras whose union is dense in A. In this paper, a classification of C*-dynamical systems of this form up to equivariant isomorphism is presented. Two pictures of the actions are given, one in terms of a modified Bratteli diagram/pathspace construction, and one in terms of a modified K0 functor.

Bratteli diagrams via the De Concini–Procesi theorem

2020 ◽  
pp. 2050073
Author(s):  
Daniele Mundici
Keyword(s):  
Large Class ◽  
Word Problem ◽  
Bratteli Diagram ◽  
Von Neumann ◽  
Generators And Relations ◽  
The Core ◽  
Bratteli Diagrams ◽  
Group Presentations ◽  
Almost All ◽  
Af Algebras

An AF algebra [Formula: see text] is said to be an AF[Formula: see text] algebra if the Murray–von Neumann order of its projections is a lattice. Many, if not most, of the interesting classes of AF algebras existing in the literature are AF[Formula: see text] algebras. We construct an algorithm which, on input a finite presentation (by generators and relations) of the Elliott semigroup of an AF[Formula: see text] algebra [Formula: see text], generates a Bratteli diagram of [Formula: see text] We generalize this result to the case when [Formula: see text] has an infinite presentation with a decidable word problem, in the sense of the classical theory of recursive group presentations. Applications are given to a large class of AF algebras, including almost all AF algebras whose Bratteli diagram is explicitly described in the literature. The core of our main algorithms is a combinatorial-polyhedral version of the De Concini–Procesi theorem on the elimination of points of indeterminacy in toric varieties.

THE KLEIN-4 DIAGRAM ALGEBRAS

2008 ◽  
Vol 07 (02) ◽  
pp. 231-262
Author(s):  
M. PARVATHI ◽  
B. SIVAKUMAR
Keyword(s):  
Bratteli Diagram ◽  
Permutation Representation ◽  
Signed Permutation ◽  
Permutation Module ◽  
New Class ◽  
Diagram Algebras ◽  
Irreducible Modules ◽  
Centralizer Algebras ◽  
Partition Algebras ◽  
Tensor Powers

In this paper we study a new class of diagram algebras, the Klein-4 diagram algebras denoted by Rk(n). These algebras are the centralizer algebras of the group Gn := (ℤ2 × ℤ2)≀Sn acting on V⊗k, where V is the signed permutation module for Gn These algebras have been realized as subalgebras of the extended G-vertex colored partition algebras introduced by Parvathi and Kennedy in [7]. In this paper we give a combinatorial rule for the decomposition of the tensor powers of the signed permutation representation of Gn by explicitly constructing the basis for the irreducible modules. In the process we also give the basis for the irreducible modules appearing in the decomposition of V⊗k in [5]. We then use this rule to describe the Bratteli diagram of Klein-4 diagram algebras.

scholarly journals On the eigenvalues of finite rank Bratteli–Vershik dynamical systems

2009 ◽  
Vol 30 (3) ◽  
pp. 639-664 ◽  
Author(s):  
XAVIER BRESSAUD ◽  
FABIEN DURAND ◽  
ALEJANDRO MAASS
Keyword(s):  
Finite Rank ◽  
Additive Group ◽  
Bratteli Diagram ◽  
Necessary Condition ◽  
Incidence Matrices ◽  
Bounded Number ◽  
Stable Space ◽  
To Come ◽  
Cantor Minimal Systems ◽  
Stable Subspace

AbstractIn this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated with the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition for there to be a measurable eigenvalue. Then, we consider two families of examples, a first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally, we study Toeplitz-type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove that measurable eigenvalues are always rational but not necessarily continuous.

scholarly journals A topological isomorphism invariant for certain AF algebras

2005 ◽  
Vol 2005 (11) ◽  
pp. 1665-1673 ◽  
Author(s):  
Ryan J. Zerr
Keyword(s):  
Topological Space ◽  
Automorphism Groups ◽  
Bratteli Diagram ◽  
Topological Isomorphism ◽  
Dimension Group ◽  
Af Algebras

For certain AF algebras, a topological space is described which provides an isomorphism invariant for the algebras in this class. These AF algebras can be described in graphical terms by virtue of the existence of a certain type of Bratteli diagram, and the order-preserving automorphisms of the corresponding AF algebra's dimension group are then studied by utilizing this graph. This will also provide information about the automorphism groups of the corresponding AF algebras.

C*-algebras associated with presentations of subshifts ii. ideal structure and lambda-graph subsystems

2006 ◽  
Vol 81 (3) ◽  
pp. 369-385 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Bratteli Diagram ◽  
Ideal Structure ◽  
View Point ◽  
Bijective Correspondence ◽  
C Algebras ◽  
Graph Theoretic ◽  
Labeled Graph ◽  
Shift Transformation ◽  
Labeled Graphs ◽  
Graph System

AbstractA λ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. InDoc. Math.7 (2002) 1–30, the author constructed aC*-algebraO£associated with a λ-graph system £ from a graph theoretic view-point. If a λ-graph system comes from a finite labeled graph, the algebra becomes a Cuntz-Krieger algebra. In this paper, we prove that there is a bijective correspondence between the lattice of all saturated hereditary subsets of £ and the lattice of all ideals of the algebraO£, under a certain condition on £ called (II). As a result, the class of theC*-algebras associated with λ-graph systems under condition (II) is closed under quotients by its ideals.

scholarly journals Invariant measures on stationary Bratteli diagrams

2009 ◽  
Vol 30 (4) ◽  
pp. 973-1007 ◽  
Author(s):  
S. BEZUGLYI ◽  
J. KWIATKOWSKI ◽  
K. MEDYNETS ◽  
B. SOLOMYAK
Keyword(s):  
Dynamical Systems ◽  
Equivalence Relation ◽  
Complex Number ◽  
Incidence Matrix ◽  
Invariant Measures ◽  
Explicit Description ◽  
Bratteli Diagram ◽  
Probability Measures ◽  
Bratteli Diagrams ◽  
Substitution Dynamical Systems

AbstractWe study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we give an explicit description of all ergodic probability measures that are invariant with respect to the tail equivalence relation (or the Vershik map); these measures are completely described by the incidence matrix of the Bratteli diagram. Since such diagrams correspond to substitution dynamical systems, our description provides an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.

Sign in / Sign up

Export Citation Format

Share Document