arXiv subjects

Kengo Matsumoto

Publications and source records attributed to Kengo Matsumoto.

62 records · Page 4Linked to original sources

arXiv2007

Orbit equivalence of one-sided subshifts and the associated C^*-algebras

A $λ$-graph system ${\frak L}$ is a generalization of a finite labeled graph and presents a subshift. We will prove that the topological dynamical systems $(X_{{\frak L}_1},σ_{{\frak L}_1})$ and $(X_{{\frak L}_2},σ_{{\frak L}_2})$ for $λ$-graph systems ${\frak L}_1$ and ${\frak L}_2$ are continuously orbit equivalent if and only if there exists an isomorphism between the associated $C^*$-algebras ${\Cal O}_{{\frak L}_1}$ and ${\Cal O}_{{\frak L}_2}$ keeping their commutative $C^*$-subalgebras $C(X_{{\frak L}_1})$ and $C(X_{{\frak L}_2})$. It is also equivalent to the condition that there exists a homeomorphism from $X_{{\frak L}_1}$ to $X_{{\frak L}_2}$ intertwining their topological full inverse semigroups. In particular, one-sided subshifts $X_{Λ_1}$ and $X_{Λ_2}$ are $λ$-continuously orbit equivalent if and only if there exists an isomorphism between the associated $C^*$-algebras ${\Cal O}_{Λ_1}$ and ${\Cal O}_{Λ_2}$ keeping their commutative $C^*$-subalgebras $C(X_{Λ_1})$ and $C(X_{Λ_2})$.

math.OA↗

arXiv2007

Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras

We will prove that one-sided topological Markov shifts $(X_A,σ_A)$ and $(X_B,σ_B)$ for matrices $A$ and $B$ with entries in $\{0,1\}$ are topologically orbit equivalent if and only if there exists an isomorphism between the Cuntz-Krieger algebras ${\Cal O}_A$ and ${\Cal O}_B$ keeping their commutative $C^*$-subalgerbas $C(X_A)$ and $C(X_B)$. It is also equivalent to the condition that there exists a homeomorphism from $X_A$ to $X_B$ intertwining their topological full groups. We will also study structure of the automorphisms of ${\Cal O}_A$ keeping the commutative $C^*$-algebra $C(X_A)$.

math.OA↗

arXiv2007

Zeta functions and topological entropy of the Markov-Dyck shifts

The Markov-Dyck shifts arise from finite directed graphs. An expression for the zeta function of a Markov-Dyck shift is given. The derivation of this expression is based on a formula in Keller (G. Keller, {\it Circular codes, loop counting, and zeta-functions}, J. Combinatorial Theory {\bf 56} (1991), pp. 75--83). For a class of examples that includes the Fibonacci-Dyck shift the zeta functions and topological entropy ae determined.

math.DS↗

arXiv2007

Actions of symbolic dynamical systems on $C^$-algebras II. Simplicity of $C^$-symbolic crossed products and some examples

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 examples such as irrational rotation Cuntz-Krieger algebras will be studied.

math.OA↗

arXiv2006

Cuntz-Krieger algebras and a generalization of Catalan numbers

We first observe that the relations of the canonical generating isometries of the Cuntz algebra ${\cal O}_N$ are naturally related to the $N$-colored Catalan numbers. For a directed graph $G$, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz-Krieger algebra ${\cal O}_{A^G}$ for the transition matrix $A^G$ of $G$. The generalized Catalan numbers $c_n^G, n=0,1,2,...$ enumerate the number of Dyck paths and oriented rooted trees for the graph $G$. Its generating functions will be studied.

math.OA↗

arXiv2006

$C^*$-algebras arising from Dyck systems of topological Markov chains

Let $A$ be an $N \times N $ irreducible matrix with entries in $\{0,1\}$. We define the topological Markov Dyck shift $D_A$ to be a nonsofic subshift consisting of the $2N$ brackets $(_1,...,(_N,)_1,...,)_N$ with both standard bracket rule and Markov chain rule coming from $A$. The subshift is regarded as a subshift defined by the canonical generators $S_1^*,..., S_N^*, S_1,..., S_N $ of the Cuntz-Krieger algebra ${\Cal O}_A$. We construct an irreducible $λ$-graph system ${{\frak L}^{Ch(D_A)}}$ that presents the subshift $D_A$ so that we have an associated simple purely infinite $C^*$-algebra ${\Cal O}_{{\frak L}^{Ch(D_A)}}$. We prove that ${\Cal O}_{{\frak L}^{Ch(D_A)}}$ is a universal unique $C^*$-algebra subject to some operator relations among $2N$ generating partial isometries. Some examples are presented such that they are not stably isomorphic to any Cuntz-Krieger algebra.

math.OA↗

arXiv2006

K-theory for the simple $C^*$-algebra of the Fibonacchi Dyck system

Let $F$ be the Fibonacci matrix $ \bigl[\begin{smallmatrix} 1 & 1 1 & 0 \\ \end{smallmatrix}\bigr] $. The Fibonacci Dyck shift is a subshsystem of the Dyck shift $D_2$ constrained by the matrix $F$. Let ${{\frak L}^{Ch(D_F)}}$ be a $λ$-graph system presenting the subshift $D_F$, that is called the Cantor horizon $λ$-graph system for $D_F$. We will study the $C^*$-algebra ${\cal O}_{{\frak L}^{Ch(D_F)}}$ associated with $ {{\frak L}^{Ch(D_F)}} $. It is simple purely infinite and generated by four partial isometries with some operator relations. We will compute the K-theory of the $C^*$-algebra. As a result, the $C^*$-algebra is simple purely infinite and not semiprojective. Hence it is not stably isomorphic to any Cuntz-Krieger algebra.

math.OA↗

arXiv2006

Textile systems on lambda-graph systems

The notions of symbolic matrix system and $λ$-graph system for a subshift are generalizations of symbolic matrix and $λ$-graph (= finite symbolic matrix) for a sofic shift respectively ([Doc. Math. 4(1999), 285-340]). M. Nasu introduced the notion of textile system for a pair of graph homomorphisms to study automorphisms and endomorphisms of topological Markov shifts ([Mem. Amer. Math. Soc. 546,114(1995)]). In this paper, we formulate textile systems on $λ$-graph systems and study automorphisms on subshifts. We will prove that for a forward automorphism $ϕ$ of a subshift $(Λ,σ)$, the automorphisms $ϕ^k σ^n, k\ge 0, n\ge 1$ can be explicitly realized as a subshift defined by certain symbolic matrix systems coming from both the strong shift equivalence representing $ϕ$ and the subshift $(Λ,σ)$. As an application of this result, if an automorphism $ϕ$ of a subshift $Λ$ is a simple automorphism, the dynamical system $(Λ, ϕ\circ σ)$ is topologically conjugate to the subshift $(Λ, σ).$

math.OA↗