Building on our previous work, we give a thorough presentation of the techniques developed for synchronizing dynamical systems in the special case of synchronizing shift spaces. Following the work of Thomsen, we give a construction of the homoclinic, the heteroclinic, and synchronizing heteroclinic C^{\ast} -algebras along with the synchronizing ideal of a shift space in terms of Bratteli diagrams. The algebras introduced in our previous work (the synchronizing ideal and synchronizing heteroclinic algebra) are discussed in detail. In the sofic shift case, these algebras are shown to be related to the C^{\ast} -algebras of its minimal left and minimal right presentations. Several specific examples are discussed to demonstrate these techniques. For the even shift, we give a complete computation of all the associated invariants. We discuss these algebras for a sofic shift that is not of almost finite type and for a number of strictly non-sofic synchronizing shifts. In particular, we discuss the rank of the K -theory of the homoclinic algebra of a shift space and its synchronizing ideal and its implications. We also give a construction for producing from any minimal shift a synchronizing shift whose set of non-synchronizing points is exactly the original minimal shift.

Bratteli diagrams with countably infinite levels exhibit a new phenomenon: they can be horizontally stationary. The incidence matrices of these horizontally stationary Bratteli diagrams are infinite banded Toeplitz matrices. In this paper, we study the fundamental properties of horizontally stationary Bratteli diagrams. In these diagrams, we provide an explicit description of ergodic tail invariant probability measures. For a certain class of horizontally stationary Bratteli diagrams, we prove that all ergodic tail invariant probability measures are extensions of measures from odometers. Additionally, we establish conditions for the existence of a continuous Vershik map on the path space of a horizontally stationary Bratteli diagram.

Abstract The aim of this paper is to study measure-theoretical rigidity and partial rigidity for classes of Cantor dynamical systems including Toeplitz systems and enumeration systems. We use Bratteli diagrams to control invariant measures that are produced in our constructions. This leads to systems with desired properties. Among other things, we show that there exists a Toeplitz system with zero entropy that is not partially measure-theoretically rigid with respect to its (unique) invariant measure. We investigate enumeration systems defined by a linear recursion, prove that all such systems are partially rigid and present an example of an enumeration system which is not measure-theoretically rigid. We construct a minimal 𝒮 {\mathcal{S}} -adic Toeplitz subshift which has countably infinitely many ergodic invariant probability measures which are rigid for the same rigidity sequence.

Hopf–Galois extensions extend the idea of principal bundles to noncommutative geometry, using Hopf algebras as symmetries. We show that the matrix embeddings in Bratteli diagrams are iterated direct sums of Hopf–Galois extensions (quantum principal bundles) for certain finite abelian groups. The corresponding strong universal connections are computed. We show that Mn(C) is a trivial quantum principle bundle for the Hopf algebra C[Zn×Zn]. We conclude with an application relating calculi on groups to calculi on matrices.

Bratteli–Vershik models have been very successfully applied to the study of various dynamical systems, in particular, in Cantor dynamics. In this paper, we study dynamics on the path spaces of generalized Bratteli diagrams that form models for non-compact Borel dynamical systems. Generalized Bratteli diagrams have countably infinite many vertices at each level; thus, the corresponding incidence matrices are also countably infinite. We emphasize differences (and similarities) between generalized and classical Bratteli diagrams. Our main results are as follows. (i) We utilize Perron–Frobenius theory for countably infinite matrices to establish criteria for the existence and uniqueness of tail-invariant path space measures (both probability and \sigma -finite). (ii) We provide criteria for the topological transitivity of the tail equivalence relation. (iii) We describe classes of stationary generalized Bratteli diagrams (hence Borel dynamical systems) that (a) do not support a probability tail-invariant measure and (b) are not uniquely ergodic with respect to the tail equivalence relation. (iv) We describe classes of generalized Bratteli diagrams which can or cannot admit a continuous Vershik map and construct a Vershik map which is a minimal homeomorphism of a (non-locally compact) Polish space. (v) We provide an application of the theory of stochastic matrices to analyze diagrams with positive recurrent incidence matrices.

Following Robert’s [J. Reine Angew. Math. 756 (2019), pp. 285–319], we study the structure of unitary groups and groups of approximately inner automorphisms of unital C ∗ C^* -algebras, taking advantage of the former being Banach-Lie groups. For a given unital C ∗ C^* -algebra A A , we provide a description of the closed normal subgroup structure of the connected component of the identity of the unitary group, denoted by U A U_A , resp. of the subgroup of approximately inner automorphisms induced by the connected component of the identity of the unitary group, denoted by V A V_A , in terms of perfect ideals, i.e. ideals admitting no characters. When the unital algebra is locally AF, we show that there is a one-to-one correspondence between closed normal subgroups of V A V_A and perfect ideals of the algebra, which can be in the separable case conveniently described using Bratteli diagrams; in particular showing that every closed normal subgroup of V A V_A is perfect. We also characterize unital C ∗ C^* -algebras A A such that U A U_A , resp. V A V_A are topologically simple, generalizing the main results of Robert [J. Reine Angew. Math. 756 (2019), pp. 285–319] from \cite{Rob19}. In the other way round, under certain conditions, we characterize simplicity of the algebra in terms of the structure of the unitary group. This in particular applies to reduced group C ∗ C^* -algebras of discrete groups and we show that when A A is a reduced group C ∗ C^* -algebra of a non-amenable countable discrete group, then A A is simple if and only if U A / T U_A/\mathbb {T} is topologically simple.

Objectives: The purpose of this work is to study 𝑝-Bratteli diagrams arising from a class of group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 for every odd prime 𝑝, to define the irreducible representations using mutually orthogonal primitive idempotents and to index the complete set of irreducible representations for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟. Method: We use primitive idempotents to define irreducible representations and systematically index the irreducible representations to establish vertices for the 𝑝-Bratteli diagram. Also, we compute the matrix units using the orthogonal primitive idempotents of the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟. Findings: The complete set of irreducible representations for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 has been successfully indexed, and matrix units for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 were determined using the orthogonal primitive idempotents. Novelty: This work provides a structured approach to constructing 𝑝-Bratteli diagrams for the group algebras related to odd primes, contributing new insights into their irreducible representations and matrix unit’s structures. Keywords: Primitive idempotent; Irreducible representation; Matrix units; Young diagram; Bratteli diagram

Abstract We introduce a generalization of immanants of matrices, using partition algebra characters in place of symmetric group characters. We prove that our immanant-like function on square matrices, which we refer to as the recombinant, agrees with the usual definition for immanants for the special case whereby the vacillating tableaux associated with the irreducible characters correspond, according to the Bratteli diagram for partition algebra representations, to the integer partition shapes for symmetric group characters. In contrast to previously studied variants and generalizations of immanants, as in Temperley–Lieb immanants and f-immanants, the sum that we use to define recombinants is indexed by a full set of partition diagrams, as opposed to permutations.

In 2010, Bezuglyi, Kwiatkowski, Medynets, and Solomyak [10] found a complete description of the set of probability ergodic tail invariant measures on the path space of a standard (classical) stationary reducible Bratteli diagram.It was shown that every distinguished eigenvalue for the incidence matrix determines a probability ergodic invariant measure.In this paper, we show that this result does not hold for stationary reducible generalized Bratteli diagrams.We consider classes of stationary and non-stationary reducible generalized Bratteli diagrams with infinitely many simple standard subdiagrams, in particular, with infinitely many odometers as subdiagrams.We characterize the sets of all probability ergodic invariant measures for such diagrams and study partial orders under which the diagrams can support a Vershik homeomorphism.

In this paper, we extend Ocneanu’s theory of connections on graphs to define a 2-category whose 0-cells are tracial Bratteli diagrams, and whose 1-cells are generalizations of unitary connections. We show that this 2-category admits an embedding into the 2-category of hyperfinite von Neumann algebras, generalizing fundamental results from subfactor theory to a 2-categorical setting.

We introduce a generalisation $LH_n$ of the ordinary Hecke algebras informed by the loop braid group $LB_n$ and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that $LH_n$ has analogues of several of these properties. In particular we %introduce consider a class of local (tensor space/functor) representations of the braid group derived from a meld of the (non-functor) Burau representation and the (functor) Deguchi {\em et al}-Kauffman--Saleur-Rittenberg representations here called Burau-Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau-Rittenberg representation extends to a loop Burau-Rittenberg representation. And this factors through $LH_n$. Let $SP_n$ denote the corresponding quotient algebra, $k$ the ground ring, and $t \in k$ the loop-Hecke parameter. We prove the following: 1) $LH_n$ is finite dimensional over a field. 2) The natural inclusion $LB_n \rightarrow LB_{n+1}$ passes to an inclusion $SP_n \rightarrow SP_{n+1}$. 3) Over $k=\mathbb{C}$, $SP_n / rad $ is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. 4) We determine the other fundamental invariants of $SP_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. 5) The structure of $SP_n $ is independent of the parameter $t$, except for $t= 1$. \item For $t^2 \neq 1$ then $LH_n \cong SP_n$ at least up to rank$n=7$ (for $t=-1$ they are not isomorphic for $n>2$; for $t=1$ they are not isomorphic for $n>1$). Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.

Lifting Bratteli diagrams between Krajewski diagrams: Spectral triples, spectral actions, and AF algebras

ABSTRACT A bounded topological speedup of a Cantor minimal system is a minimal system , where for some bounded function , or any system topologically conjugate to such an . Assuming the system is represented by a properly ordered Bratteli diagram , we provide a method for constructing a new, perfectly ordered Bratteli diagram that represents the sped-up system . The diagram relates back to in a manner that enables us to see how certain dynamical properties are preserved under speedup. As an application, in the case that is a substitution minimal system, we show how to use to write an explicit substitution rule that generates the sped-up system , answering an open question from [L. Alvin, D.D. Ash, and N.S. Ormes, Bounded topological speedups, Dyn. Syst. 33(2) (2018), pp. 303–331.].

On the monotone C⁎-algebra

We consider a tower of generalized rook monoid algebras over the field \mathbb{C} of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys–Murphy elements for generalized rook monoid algebras. Over an algebraically closed field \Bbbk of positive characteristic p , utilizing Jucys–Murphy elements of rook monoid algebras, for 0\leq i\leq p-1 we define the corresponding i -restriction and i -induction functors along with two extra functors. On the direct sum \mathcal{G}_{\mathbb{C}} of the Grothendieck groups of module categories over rook monoid algebras over \Bbbk , these functors induce an action of the tensor product of the universal enveloping algebra U(\widehat{\mathfrak{sl}}_p(\mathbb{C})) and the monoid algebra \mathbb{C}[\mathcal{B}] of the bicyclic monoid \mathcal{B} . Furthermore, we prove that \mathcal{G}_{\mathbb{C}} is isomorphic to the tensor product of the basic representation of U(\widehat{\mathfrak{sl}}_{p}(\mathbb{C})) and the unique infinite-dimensional simple module over \mathbb{C}[\mathcal{B}] , and also exhibit that \mathcal{G}_{\mathbb{C}} is a bialgebra. Under some natural restrictions on the characteristic of \Bbbk , we outline the corresponding result for generalized rook monoids.

The past decade has seen a flourishing of advances in harmonic analysis of graphs. They lie at the crossroads of graph theory and such analytical tools as graph Laplacians, Markov processes and associated boundaries, analysis of path-space, harmonic

The infinite product of matrices with integer entries, known as a modified Glimm–Bratteli symbol n, is a new, sufficiently simple, and very powerful tool for the characterization of approximately finite-dimensional (AF) algebras. This symbol provides a convenient algebraic representation of the Bratteli diagram for AF algebras in the same way as was previously performed by J. Glimm for more simple uniformly hyperfinite (UHF) algebras. We apply this symbol to characterize integrodifferential algebras. The integrodifferential algebra FN,M is the C∗-algebra generated by the following operators acting on L2([0,1)N→CM): (1) operators of multiplication by bounded matrix-valued functions, (2) finite-difference operators, and (3) integral operators. Most of the operators and their approximations studying in physics belong to these algebras. We give a complete characterization of FN,M. In particular, we show that FN,M does not depend on M, but depends on N. At the same time, it is known that differential algebras HN,M, generated by the operators (1) and (2) only, do not depend on both dimensions N and M; they are all ∗-isomorphic to the universal UHF algebra. We explicitly compute the Glimm–Bratteli symbols (for HN,M, it was already computed earlier) which completely characterize the corresponding AF algebras. This symbol n is an infinite product of matrices with nonnegative integer entries. Roughly speaking, all the symmetries appearing in the approximation of complex infinite-dimensional integrodifferential and differential algebras by finite-dimensional ones are coded by a product of integer matrices.

A λ-graph bisystem consists of a pair of two labelled Bratteli diagrams, that presents a two-sided subshift . We will construct a compact totally disconnected metric space consisting of tilings of a two-dimensional half plane from a λ-graph bisystem. The tiling space has a certain AF-equivalence relation written with a natural shift homeomorphism coming from the shift homeomorphism on the subshift . The equivalence relation yields an AF-algebra with an automorphism induced by . We will study invariance of the étale equivalence relation , the groupoid and the groupoid -algebras , under topological conjugacy of the presenting two-sided subshifts.

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a.~ topological full groups). We determine that any piecewise full group that is locally compact in the compact-open topology on the group of self-homeomorphisms of the Cantor set must be uniformly discrete, in a precise sense that we introduce here. Uniformly discrete groups of self-homeomorphisms of the Cantor set are in particular countable, locally finite, residually finite and discrete in the compact-open topology. The resulting piecewise full groups form a subclass of the ample groups introduced by Krieger. We determine the structure of these groups by means of their Bratteli diagrams and associated dimension ranges ($K_0$ groups). We show through an example that not all uniformly discrete piecewise full groups are subgroups of the ``obvious'' ones, namely, piecewise full groups of finite groups.

In this paper, we define some Markov chains associated with Vershik maps on Bratteli diagrams. We study probabilistic and spectral properties of their transition operators and we prove that the spectra of these operators are connected to Julia sets in higher dimensions. We also study topological properties of these spectra.