arXiv subjects
Publications and source records attributed to Kengo Matsumoto.
In this paper, we prove that the continuous orbit equivalence class of a one-sided topological Markov shift contains a one-sided topological Markov shift whose topological entropy is greater than an arbitrary prescribed positive real number, and also contains a one-sided topological Markov shift whose topological entropy is less than an arbitrary prescribed positive real number.
The reciprocality means a duality in Kirchberg algebras between K-theory groups and strong extension groups. In the paper, we will find a certain class of unital simple Exel--Laca algebras for which the reciprocal duals are simple Cuntz--Krieger algebras in terms of the underlying infinite matrices. In our procedure to obtain simple Cuntz--Krieger algebras from Exel--Laca algebras, we compute the strong extension groups for Exel--Laca algebras belonging to the class.
Reciprocality in Kirchberg algebras with finitely generated K-groups is regarded as a K-theoretic duality through K-groups and strong extension groups. We will prove that the reciprocal Kirchberg algebra has a universal property with respect to some generating C*-subalgebra and a family of generating partial isometries. By using the universal property, we will prove that there exists an aperiodic ergodic automorphism on an arbitrary unital Kirchberg algebra with finitely generated K-groups, which has a unique invariant state. The state is pure.
Combining the theory of extensions of C*-algebras and the Pimsner construction, we show that every countable infinite discrete group admits an ergodic action on arbitrary unital Kirchberg algebra. In the proof, we give a Pimsner construction realizing many unital subalgebras of a given unital Kirchberg algebra as the fixed point algebras of single automorphisms. Furthermore, for amenable infinite discrete groups, we show that every point-wise outer action on arbitrary unital Kirchberg algebra has an ergodic cocycle perturbation with the help of Gabe--Szabó's theorem and Baum--Connes' conjecture.
Reciprocality in Kirchberg algebras is a duality between strong extension groups and K-theory groups. We describe a construction of the reciprocal dual algebra $\widehat{\mathcal{A}}$ for a Kirchberg algebra $\mathcal{A}$ with finitely generated K-groups via K-theoretic duality for extensions. In particular, we may concretely realize the reciprocal algebra $\widehat{\mathcal{O}}_A$ for simple Cuntz--Krieger algebras $\mathcal{O}_A$. As a result, the algebra $\widehat{\mathcal{O}}_A$ is realized as a unital simple purely infinite universal $C^*$-algebra generated by a family of partial isometries subject to certain operator relations. We will also study gauge actions on the reciprocal algebra $\widehat{\mathcal{O}}_A$ and prove that there exists an isomorphism between the fundamental groups $π_1({\operatorname{Aut}}({\mathcal{O}}_A))$ and $π_1({\operatorname{Aut}}(\widehat{\mathcal{O}}_A))$ preserving their gauge actions.
We introduce a hierarchy for unital Kirchberg algebras with finitely generated K-groups by which the first and second homotopy groups of the automorphism groups serve as a complete invariant of classification. We also introduce an invariant called the total extension group which is the direct sum of the strong and weak extension groups. In the case of unital Kirchberg algebras with finitely generated K-groups, the total extension group gives a complete invariant and provides a useful tool to classify the Cuntz--Krieger algebras.
A $λ$-graph system is a labeled Bratteli diagram with certain additional structure, which presents a subshift. The class of the $C^*$-algebras $\mathcal{O}_{\frak L}$ associated with the $λ$-graph systems is a generalized class of the class of Cuntz--Krieger algebras. In this paper, we will compute the strong extension groups $\operatorname{Ext}_{\operatorname{s}}(\mathcal{O}_{\frak L})$ for the $C^*$-algebras associated with $λ$-graph systems ${\frak L}$ and study their relation with the weak extension group $\operatorname{Ext}_{\operatorname{w}}(\mathcal{O}_{\frak L})$.
In this paper, we first present the homotopy groups of the automorphism groups of Cuntz--Krieger algebras in terms of the underlying matrices of the Cuntz--Krieger algebras. We also show that the homotopy groups are complete invariants of the isomorphism class of the Cuntz--Krieger algebras. As a result, the isomorphism type of Cuntz--Krieger algebras are completely characterized by the group structure of the weak and strong extension groups.
In this paper, we study the strong extension groups of Cuntz--Krieger algebras, and present a formula to compute the groups. We also detect the position of the Toeplitz extension of a Cuntz--Krieger algebra in the strong extension group and in the weak extension group to see that the weak extension group with the position of the Toeplitz extension is a complete invariant of the isomorphism class of the Cuntz--Krieger algebra associated with its transposed matrix.
We introduce the notion of K-theoretic duality for extensions of separable unital nuclear $C^*$-algebras by using K-homology long exact sequence and cyclic six term exact sequence for K-theory groups of extensions. We then prove that the Toeplitz extension $\mathcal{T}_A$ of a Cuntz-Krieger algebra $\mathcal{O}_A$ is the K-theoretic dual of the Toeplitz extension $\mathcal{T}_{A^t}$ of the Cuntz-Krieger algebra $\mathcal{O}_{A^t}$ for the transposed matrix $A^t$ of $A$. A pair of isomorphic Cuntz--Krieger algebras $\mathcal{O}_A$ and $\mathcal{O}_B$ does not necessarily yield the isomorphic pair of $\mathcal{O}_{A^t}$ and $\mathcal{O}_{B^t}.$ However, as an application, we may show that two Toeplitz algebras $\mathcal{T}_A$ and $\mathcal{T}_B$ are isomorphic as $C^*$-algebras if and only if the Toeplitz algebras $\mathcal{T}_{A^t}$ and $\mathcal{T}_{B^t}$ of their transposed matrices are isomorphic.
We study asymptotic continuous orbit equivalence of Smale spaces. We prove that two irreducible Smale spaces are flip conjugate if and only if there exists a periodic point preserving homeomorphism giving an asymptotic continuous orbit equivalence between them. We introduce a notion of asymptotic topological conjugacy and asymptotic flip conjugacy in Smale spaces and characterize them in terms of the associated Ruelle algebras with dual actioons. We finally characterize the flip conjugacy classes of irreducible two-sided topological Markov shifts in terms of the associated Ruelle algebras with its $C^*$-subalgebras.
The class of normal subshifts includes irreducible infinite topological Markov shifts, irreducible infinite sofic shifts, synchronized systems, Dyck shifts, $β$-shifts, substitution minimal shifts, and so on. We will characterize one-sided topological conjugacy classes of normal subshifts in terms of the associated $C^*$-algebras and its gauge actions with potentials.
In this paper, we study a family of $C^*$-subalgebras defined by fixed points of generalized gauge actions of a Cuntz-Krieger algebra, by introducing a family of étale groupoids whose associated $C^*$-algebras are these $C^*$-subalgebras. We know that topological conjugacy classes of one-sided topological Markov shifts are characterized in terms of the isomorphism classes of these étale groupoids.
We will characterize topological conjugate one-sided topological Markov shifts in terms of their subgroups of continuous full groups and subalgebras of Cuntz--Krieger algebras.
We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.
We study the $C^*$-algebras of the étale groupoids defined by the asymptotic equivalence relations for hyperbolic automorphisms on the two-dimensional torus. The algebras are proved to be four-dimensional non-commutative tori by an explicit numerical computation. The ranges of the unique tracial states of its $K_0$-groups of the $C^*$-algebras are described in terms of the hyperbolic matrices of the automorphisms on the torus.
We introduce a notion of coded equivalence in one-sided topological Markov shifts. The notion is inspired by coding theory. One-sided topological conjugacy implies coded equivalence. We will show that coded equivalence implies continuous orbit equivalence of one-sided topological Markov shifts.
We will characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz--Krieger algebras and its gauge actions with potentials.