Nuno Martins, Ricardo Severino, J. Sousa Ramos
We compute the $K$-groups for the Cuntz-Krieger algebras $\mathcal{O}_{A_{\mathcal{K}(f_μ)}}$, where $A_{\mathcal{K}(f_μ)}$ is the Markov transition matrix arising from the \textit{kneading sequence }$\mathcal{K} (f_μ)$ of the one-parameter family of real quadratic maps $f_μ$.
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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 characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz--Krieger algebras and its gauge actions with potentials.
For an irreducible non-permutation matrix $A$, the triplet $({\mathcal{O}_A},{\mathcal{D}_A},ρ^A)$ for the Cuntz-Krieger algebra ${\mathcal{O}_A}$, its canonical maximal abelian $C^*$-subalgebra ${\mathcal{D}_A}$, and its gauge action $ρ^A$ is called the Cuntz-Krieger triplet. We introduce a notion of strong Morita equivalence in the Cuntz-Krieger triplets, and prove that two Cuntz-Krieger triplets $({\mathcal{O}_A},{\mathcal{D}_A},ρ^A)$ and $({\mathcal{O}_B},{\mathcal{D}_B},ρ^B)$ are strong Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. We also show that the general…
We will study torus actions on Cuntz--Krieger algebras trivially acting on its canonical maximal abelian $C^*$-subalgebras from the view points of continuous orbit equivalence of one-sided topological Markov shifts and flow equivalence of two-sided topological Markov shifts.
We prove that if two nonnegative matrices are strong shift equivalent, the associated stable Cuntz-Krieger algebras with generalized gauge actions are conjugate. The proof is done by a purely functional analytic method and based on constructing imprimitivity bimodule from bipartite directed graphs through strong shift equivalent matrices, so that we may clarify K-theoretic behavior of the stable conjugacy between the associated stable Cuntz-Krieger algebras. We also examine our machinery for the matrices obtained by state splitting graphs, so that topological conjugacy of the topological Marko…
We study KMS states for gauge actions with potential functions on Cuntz--Krieger algebras whose underlying one-sided topological Markov shifts are continuous orbit equivalent. As a result, we have a certain relationship between topological entropy of continuous orbit equivalent one-sided topological Markov shifts.
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.
In this paper we extend work of Kawamura, see kawamura, for Cuntz-Krieger algebras O_A for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of O_A. We use these representations to describe the Perron-Frobenius operator, associated to an nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples.
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 $C(X_A)$.
We study relations between $(H,β)$--KMS states on Cuntz--Krieger algebras and the dual of the Perron--Frobenius operator $\mathcal{L}_{-βH}^{*}$. Generalising the well--studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one--one correspondence between $(H,β)$--KMS states and eigenmeasures of $\mathcal{L}_{-βH}^{*}$ for the eigenvalue 1. We then consider representations of Cuntz--Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are $\ast$--is…
Let A,B be square irreducible matrices with entries in {0,1}. We will show that if the one-sided topological Markov shifts (X_A,σ_A) and (X_B,σ_B) are continuously orbit equivalent, then the two-sided topological Markov shifts (\bar X_A,\barσ_A) and (\bar X_B,\barσ_B) are flow equivalent, and hence det(id-A)=det(id-B). As a result, the one-sided topological Markov shifts (X_A,σ_A) and (X_B,σ_B) are continuously orbit equivalent if and only if the Cuntz-Krieger algebras O_A and O_B are isomorphic and det(id-A)=det(id-B).
In this paper, we will introduce notions of relative version of imprimitivity bimodules and relative version of strong Morita equivalence for pairs of $C^*$-algebras $(\mathcal{A}, \mathcal{D})$ such that $\mathcal{D}$ is a $C^*$-subalgebra of $\mathcal{A}$ with certain conditions. We will then prove that two pairs $(\mathcal{A}_1, \mathcal{D}_1)$ and $(\mathcal{A}_2, \mathcal{D}_2)$ are relatively Morita equivalent if and only if their relative stabilizations are isomorphic. In particularly, for two pairs $(\mathcal{O}_A, \mathcal{D}_A)$ and $(\mathcal{O}_B, \mathcal{D}_B)$ of Cuntz--Krieger …
We define a notion of dynamical pressure at a self-adjoint element for a contractive completely positive self-map of an exact C*-algebra which adopts Voiculescu's approximation approach to noncommutative entropy and extends the Voiculescu-Brown topological entropy and Neshveyev-Stormer unital-nuclear pressure. A variational inequality bounding the pressure below by the free energies with respect to the Sauvageot-Thouvenot entropy is established in two stages via the introduction of a local state approximation entropy, whose associated free energies function as an intermediate term. Pimsner C*-…
We study a new class of representations of the Cuntz-Krieger algebras $\mathcal O_A$ constructed by semibranching function systems, naturally related to stationary Bratteli diagrams. The notion of isomorphic semibranching function systems is defined and studied. We show that any isomorphism of such systems implies the equivalence of the corresponding representations of Cuntz-Krieger algebra $\mathcal O_A$. In particular, we show that equivalent measures generate equivalent representations of $\mathcal O_A$. We use Markov measures which are defined on the path space of stationary Bratteli diagr…
Five shades of KMS: Statistical properties in the spectral geometry of Cuntz--Krieger algebras
本文研究拓扑马尔可夫链和Cuntz-Krieger代数的非交换几何中的谱不变量，通过log-Laplacian构建的谱三元组证明了统计特征值渐近、局部热迹渐近、局部Weyl定律以及Connes迹定理的类比，所有局部渐近由Cuntz-Krieger代数上规范作用的Kubo-Martin-Schwinger态控制。
Given a zero-one matrix A we consider certain one-parameter groups of automorphisms of the Cuntz-Krieger algebra O_A, generalizing the usual gauge group, and depending on a positive continuous function H defined on the Markov space Σ_A. The main result consists of an application of Ruelle's Perron-Frobenius Theorem to show that these automorphism groups admit a single KMS state.
It is shown that a C*-algebra generated by any faithful covariant representation of a Hilbert bimodule X is canonically isomorphic to the crossed product associated to X provided that Rieffel's induced representation functor X-ind is topologically free. It is discussed how this result could be applied to universal C*-algebras generated by relations with a circle gauge action. In particular, it leads to generalizations of isomorphism theorems for various crossed products, and is shown to be equivalent to Cuntz-Krieger uniqueness theorem for finite graph C*-algebras (on that occasion an intrigui…
Heat operators and isometry groups of Cuntz-Krieger algebras
本文利用谱非交换几何为拓扑马尔可夫链和Cuntz-Krieger代数构造热半群与谱三元组，穷尽奇K-同调，并显式计算Cuntz代数的热算子，进而刻画等距群并证明Voiculescu非交换拓扑熵在等距上消失。
Shift equivalences through the lens of Cuntz-Krieger algebras
受 Williams 问题启发，本文通过研究图 C*-代数引入三种等价关系，在仅假设 SE 的条件下提供阻碍 SSE 的新方法，并据此测量 SSE 与 RE 的差异以反驳前人证明。
In this paper, we will study presentations of the continuous full group $Γ_A$ of a one-sided topological Markov shift $(X_A,σ_A)$ for an irreducible matrix $A$ with entries in $\{0,1\}$ as a generalization of Higman-Thompson groups $V_N, 1<N \in {\mathbb{N}}$. We will show that the group $Γ_A$ can be represented as a group $Γ_A^{\operatorname{tab}}$ of matrices, called $A$-adic tables, with entries in admissible words of the shift space $X_A$, and a group $Γ_A^{\operatorname{PL}}$ of right continuous piecewise linear functions, called $A$-adic PL functions, on $[0,1]$ with finite singularities…
Given a directed graph E, we construct for each real number l a quiver whose vertex space is the topological realisation of E, and whose edges are directed paths of length l in the vertex space. These quivers are not topological graphs in the sense of Katsura, nor topological quivers in the sense of Muhly and Tomforde. We prove that when l = 1 and E is finite, the infinite-path space of the associated quiver is homeomorphic to the suspension of the one-sided shift of E. We call this quiver the suspension of E. We associate both a Toeplitz algebra and a Cuntz-Krieger algebra to each of the quiv…
We present a class of subshifts $Z_N, N = 1,2,...$ whose associated $C^*$-algebras ${\cal O}_{Z_N}$ are simple, purely infinite and not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The class of the subshifts is the first examples whose associated $C^*$-algebras are not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The subshifts $Z_N$ are coded systems whose languages are context free. We compute the topological entropy for the subshifts and show that KMS-state for gauge action on the associated $C^*$-algebra ${\cal O}_{Z_N}$ exists if and only if …
We show how the fine structure in shift-tail equivalence, appearing in the noncommutative geometry of Cuntz-Krieger algebras developed by the first two authors, has an analogue in a wide range of other Cuntz-Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz-Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz- and Cuntz-Krieger alge…
We show that for a sofic shift Lambda, Matsumoto's C*-algebra O_Lambda is isomorphic to the Cuntz-Krieger algebra of the left Krieger cover graph of Lambda.
By using C*-correspondences and Cuntz-Pimsner algebras, we associate to every subshift (also called a shift space) $X$ a C*-algebra $O_X$, which is a generalization of the Cuntz-Krieger algebras. We show that $O_X$ is the universal C*-algebra generated by partial isometries satisfying relations given by $X$. We also show that $O_X$ is a one-sided conjugacy invariant of $X$.
Let O_{Lambda} be a higher rank graph C*-algebra of rank r. For every tuple p of non-negative integers there is a canonical completely positive map Phi^p on O_{Lambda} and a subshift T^p on the path space X of the graph. We show that ht(Phi^p)=h(T^p), where ht is Voiculescu's approximation entropy and h the classical topological entropy. For a higher rank Cuntz-Krieger algebra O_M we obtain ht(Phi^p)= log r(M_1^{p_1}M_2^{p_2} ... M_r^{p_r}), r being the spectral radius. This generalises Boca and Goldstein's result for Cuntz-Krieger algebras.
We compute the K-groups of C^*-algebras arising from one-dimensional generalized solenoids. The results show that Ruelle algebras from one-dimensional generalized solenoids are one-dimensional generalizations of Cuntz-Krieger algebras.
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 …
In this paper, we introduce a $C^{\ast}$-algebra associated with a proper primitive substitution. We show that the $C^{\ast}$-algebra is simple and purely infinite and contains the associated Cuntz-Krieger algebra and the crossed product $C^{\ast}$-algebra of the corresponding Cantor minimal system. We calculate the $K$-groups.
We describe the flows of weights for von Neumann algebras arising from KMS states for the gauge actions of the Cuntz-Pimsner algebras in terms of Poisson (tail) boundaries.
We study dynamics of the Lattès maps in the complex plane in terms of the Cuntz-Krieger algebras associated to the endomorphisms of the non-commutative tori. In particular, it is shown that iterations of the Lattès maps can be reduced to the dynamics of the subshifts of finite type. Using such a reduction, we calculate the zeta function of the Lattès maps.
Boundary actions of outer automorphism groups of Thompson-like groups
本文研究了索恩-汤普森群的外自同构群的边界作用，并证明这些群是C*-简单。
Higher-rank graphs (or $k$-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz-Krieger $C^*$-algebras of Robertson and Steger. Here we consider a family of finite 2-graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others. We analyse the $C^*$-algebras of these 2-graphs, find criteria under which they are simple and purely infinite, and compute their $K$-theory. We find examples whose $C^*$-algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic …
We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z. Consequently, results of Eilers, Restorff, Ruiz and Sørensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type, and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).
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