The K-theory of Cuntz-Krieger algebras for infinite matrices

Topological Entropy for the Canonical Endomorphism of Cuntz-Krieger Algebras

It is shown that Voiculescu's toplogical entropy for the canonical endomorphism of a simple Cuntz-Krieger algebra O_A equals the logarithm of the spectral radius of A.

A Cuntz--Krieger uniqueness theorem for semigraph $C^*$-algebras

Higher rank semigraph algebras are introduced by mixing concepts of ultragraph algebras and higher rank graph algebras. This yields a kind of higher rank generalisation of ultragraph algebras. We prove Cuntz--Krieger uniqueness theorems for cancelling semigraph algebras and aperiodic full semigraph algebras.

Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz-Krieger algebras

We introduce a notion of uniformly continuous orbit equivalence as a subequivalence relation of continuous orbit equivalence of one-sided topological Markov shifts. It is described in terms of gauge actions on the associated Cuntz-Krieger algebras and continuous full groups of the Markov shifts.

Classification and realizations of type III factor representations of Cuntz-Krieger algebras associated with quasi-free states

We completely classify type III factor representations of Cuntz-Krieger algebras associated with quasi-free states up to unitary equivalence. Furthermore, we realize these representations on concrete Hilbert spaces without using GNS construction. Free groups and their type ${\rm II}_{1}$ factor representations are used in these realizations.

Tensor products of type III factor representations of Cuntz-Krieger algebras

We introduced a non-symmetric tensor product of any two states or any two representations of Cuntz-Krieger algebras associated with a certain non-cocommutative comultiplication in previous our work. In this paper, we show that a certain set of KMS states is closed with respect to the tensor product. From this, we obtain formulae of tensor product of type {\rm III} factor representations of Cuntz-Krieger algebras which is different from results of the tensor product of factors of type {\rm III}.

The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources

We catalogue the primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources. Each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph; the primitive ideals in the Cuntz-Krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group. The Cuntz-Krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic.

退相量子图的局域量子Cuntz–Krieger代数

Local quantum Cuntz--Krieger algebras of dephasing quantum graphs

本文针对有限量子图的局域量子Cuntz–Krieger族建立结构定理，将其应用于Mₙ上的量子退相图，得到Oₙ与该代数的显式同构及Oₙ在对应代数中的单位嵌入。

On certain properties of Cuntz--Krieger type algebras

The note presents a further study of the class of Cuntz--Krieger type algebras. A necessary and sufficient condition is identified that ensures that the algebra is purely infinite, the ideal structure is studied, % and applied to semigraph algebras, and nuclearity is proved by presenting the algebra as a crossed product of an AF-algebra by an abelian group. The results are applied to examples of Cuntz--Krieger type algebras, such as higher rank semigraph $C^*$-algebras and higher rank Exel-Laca algebras.

Skew products of finitely aligned left cancellative small categories and Cuntz-Krieger algebras

Given a group cocycle on a finitely aligned left cancellative small category (LCSC) we investigate the associated skew product category and its Cuntz-Krieger algebra, which we describe as the crossed product of the Cuntz-Krieger algebra of the original category by an induced coaction of the group. We use our results to study Cuntz-Krieger algebras arising from free actions of groups on finitely aligned LCSC's, and to construct coactions of groups on Exel-Pardo algebras. Finally we discuss the universal group of a small category and connectedness of skew product categories.

Strong classification of purely infinite Cuntz-Krieger algebras

In 2006, Restorff completed the classification of all Cuntz-Krieger algebras with finitely many ideals (i.e., those that are purely infinite) up to stable isomorphism. He left open the questions concerning strong classification up to stable isomorphism and unital classification. In this paper, we address both questions. We show that any isomorphism between the reduced filtered K-theory of two Cuntz-Krieger algebras with finitely many ideals lifts to a *-isomorphism between the stabilized Cuntz-Krieger algebras. As a result, we also obtain strong unital classification.

一类Exel-Laca代数与Cuntz-Krieger代数互为对偶

A class of Exel--Laca algebras reciprocal to Cuntz--Krieger algebras

研究一类单生成简单Exel-Laca代数，其对偶为简单Cuntz-Krieger代数，通过无限矩阵构造，并计算强扩展群。

Realising the Toeplitz algebra of a higher-rank graph as a Cuntz-Krieger algebra

For a row-finite higher-rank graph Λ, we construct a higher-rank graph TΛ such that the Toeplitz algebra of Λ is isomorphic to the Cuntz-Krieger algebra of TΛ. We then prove that the higher-rank graph TΛ is always aperiodic and use this fact to give another proof of a uniqueness theorem for the Toeplitz algebras of higher-rank graphs.

Twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs

To each finitely aligned higher-rank graph $Λ$ and each $\mathbb{T}$-valued 2-cocycle on $Λ$, we associate a family of twisted relative Cuntz-Krieger algebras. We show that each of these algebras carries a gauge action, and prove a gauge-invariant uniqueness theorem. We describe an isomorphism between the fixed point algebras for the gauge actions on the twisted and untwisted relative Cuntz-Krieger algebras. We show that the quotient of a twisted relative Cuntz-Krieger algebra by a gauge-invariant ideal is canonically isomorphic to a twisted relative Cuntz-Krieger algebra associated to a subgr…

Cuntz-Krieger type uniqueness theorem for topological higher-rank graph C*-algebras

We study Sims-Yeend's product system C*-algebras and topological higher-rank graph C*-algebras by Yeend. We give a relation between Katsura's Cuntz-Pimsner covariance and Sims-Yeend's one by a direct approach and an explicit form of the core of product system C*-algebras. Finally, we prove Cuntz-Krieger type uniqueness theorem for topological higher-rank graph C*-algebras under certain aperiodic condition.

量子 Cuntz-Krieger 代数上的 KMS 态

KMS states on quantum Cuntz-Krieger algebras

本文研究量子图相关的局部量子 Cuntz-Krieger 代数上的 KMS 态，利用与 Cuntz-Pimsner 代数的同构，将 KMS 态判据转化为关于量子邻接算子的命题，并实现了规范作用下 KMS 态的完全分类。

Imprimitivity bimodules of Cuntz--Krieger algebras and strong shift equivalences of matrices

In this paper, we will introduce a notion of basis related Morita equivalence in the Cuntz--Krieger algebras $({\mathcal{O}_A}, \{S_a\}_{a \in E_A})$ with the canonical right finite basis $\{S_a\}_{a \in E_A}$ as Hilbert $C^*$-bimodule, and prove that two nonnegative irreducible matrices $A$ and $B$ are elementary equivalent, that is, $A = CD, B = DC$ for some nonnegative rectangular matrices $C, D$, if and only if the Cuntz--Krieger algebras $({\mathcal{O}_A}, \{S_a\}_{a \in E_A})$ and $({\mathcal{O}_B}, \{ S_b\}_{b\in E_B})$ with the canonical right finite bases are basis relatedly elementar…

Permutative representations of the Cuntz-Krieger algebras

We generalize permutative representations of the Cuntz algebras for the \cka\ $\coa$ for any $A$. We characterize cyclic permutative representations by notions of cycle and chain, and show their existence and uniqueness. We show necessary and sufficient conditions for their irreducibility and equivalence. In consequence, we have a complete classification of permutative representations of $\coa$. Furthermore we show decomposition formulae and that the uniqueness of irreducible decomposition holds for permutative representation.

Extensions of Cuntz-Krieger algebras

We show that a C*-algebra "looking like" a Cuntz-Krieger algebra is a Cuntz-Krieger algebra. This implies that, in an appropriate sense, the class of Cuntz-Krieger algebras is closed under extensions of real rank zero.

Cuntz-Krieger-Pimsner Algebras Associated with Amalgamated Free Product Groups

We give a construction of a nuclear $C^\ast$-algebra associated with an amalgamated free product of groups, generalizing Spielberg's construction of a certain Cuntz-Krieger algebra associated with a finitely generated free product of cyclic groups. Our nuclear $C^\ast$-algebras can be identified with certain Cuntz-Krieger-Pimsner algebras. We will also show that our algebras can be obtained by the crossed product construction of the canonical actions on the hyperbolic boundaries, which proves a special case of Adams' result about amenability of the boundary action for hyperbolic groups. We wil…

Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras

To an $r$-dimensional subshift of finite type satisfying certain special properties we associate a $C^*$-algebra $\cA$. This algebra is a higher rank version of a Cuntz-Krieger algebra. In particular, it is simple, purely infinite and nuclear. We study an example: if $\G$ is a group acting freely on the vertices of an $\wt A_2$ building, with finitely many orbits, and if $Ω$ is the boundary of that building, then $C(\Om)\rtimes \G$ is the algebra associated to a certain two dimensional subshift.

The universal Boolean inverse semigroup presented by the abstract Cuntz-Krieger relations

This paper is a contribution to the theory of what might be termed $0$-dimensional non-commutative spaces. We prove that associated with each inverse semigroup $S$ is a Boolean inverse semigroup presented by the abstract versions of the Cuntz-Krieger relations. We call this Boolean inverse semigroup the Exel completion of $S$ and show that it arises from Exel's tight groupoid under non-commutative Stone duality.

On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras

We show that an outer action of a finite abelian group on a simple Cuntz-Krieger algebra is strongly approximately inner in the sense of Izumi if the action is given by diagonal quasi-free automorphisms and the associated matrix is aperiodic. This is achieved by an approximate cohomology vanishing-type argument for the canonical shift restricted to the relative commutant of the set of domain projections of the canonical generating isometries in the fixed point algebra.

相对广义布尔动力系统的C*-代数的Cuntz--Krieger唯一性定理

A Cuntz--Krieger Uniqueness theorem for C*-algebras of relative generalized Boolean dynamical systems

本文证明了相对广义布尔动力系统C*-代数的Cuntz--Krieger唯一性定理，定义了极小性概念，并给出了C*-代数单性的充要条件。

Automorphisms of Cuntz-Krieger algebras

We prove that the natural homomorphism from Kirchberg's ideal-related KK-theory, KKE(e, e'), with one specified ideal, into Hom_Λ (\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all extensions e and e' of separable, nuclear C*-algebras in the bootstrap category N with the K-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the K_{1}-groups of the quotients being free abelian groups. This class includes all Cuntz-Krieger algebras with exactly one non-trivial ideal. Combining our results with the results o…

Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids

A one-sided shift of finite type $(X_A,σ_A)$ determines on the one hand a Cuntz-Krieger algebra $\mathcal{O}_A$ with a distinguished abelian subalgebra $\mathcal{D}_A$ and a certain completely positive map $τ_A$ on $\mathcal{O}_A$. On the other hand, $(X_A,σ_A)$ determines a groupoid $\mathcal{G}_A$ together with a certain homomorphism $ε_A$ on $\mathcal{G}_A$. We show that this data completely characterizes the one-sided conjugacy class of $X_A$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate b…

On a Cuntz-Krieger functor

We construct a covariant functor from the topological torus bundles to the so-called Cuntz-Krieger algebras; the functor maps homeomorphic bundles into the stably isomorphic Cuntz-Krieger algebras. It is shown, that the K-theory of the Cuntz-Krieger algebra encodes torsion of the first homology group of the bundle. We illustrate the result by examples.

Cuntz-Pimsner代数的Cuntz-Krieger唯一性定理

A Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner Algebras

本文引入C*-对应的条件(S)作为图的条件(L)的类比，并据此证明Cuntz-Pimsner代数的Cuntz-Krieger唯一性定理及单纯性充分条件，同时建立拓扑箭图条件(L)与对应条件(S)的等价关系。

Branching Systems and General Cuntz-Krieger Uniqueness Theorem for Ultragraph C*-algebras

We give a notion of branching systems on ultragraphs. From this we build concrete representations of ultragraph C*-algebras on the bounded linear operators of Hilbert spaces. To each branching system of an ultragraph we describe the associated Perron-Frobenius operator in terms of the induced representation. We show that every permutative representation of an ultragraph C*-algebra is unitary equivalent to a representation arising from a branching system. We give a sufficient condition on ultragraphs such that a large class of representations of the C*-algebras of these ultragraphs is permutati…

One-sided topological conjugacy of topological Markov shifts, continuous full groups and Cuntz--Krieger algebras

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.

The K-theoretical range of Cuntz-Krieger algebras

We augment Restorff's classification of purely infinite Cuntz-Krieger algebras by describing the range of his invariant on purely infinite Cuntz-Krieger algebras. We also describe its range on purely infinite graph C*-algebras with finitely many ideals, and provide 'unital' range results for purely infinite Cuntz-Krieger algebras and unital purely infinite graph C*-algebras.

Cuntz-Krieger algebras associated with Hilbert $C^*$-quad modules of commuting matrices

Let ${\cal O}_{{\cal H}^{A,B}_κ}$ be the $C^*$-algebra associated with the Hilbert $C^*$-quad module arising from commuting matrices $A,B$ with entries in $\{0,1\}$. We will show that if the associated tiling space $X_{A,B}^κ$ is transitive, the $C^*$-algebra ${\cal O}_{{\cal H}^{A,B}_κ}$ is simple and purely infinite. In particulr, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^*$-algebra ${\cal O}_{{\cal H}^{[N],[M]}_κ}$ are computed by using the Euclidean algorithm.

On one-sided topological conjugacy of topological Markov shifts and gauge actions on 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.

Boundary actions for affine buildings and higher rank Cuntz-Krieger algebras

Let $\G$ be a group of type rotating automorphisms of an affine building $\cB$ of type $\wt A_2$. If $\G$ acts freely on the vertices of $\cB$ with finitely many orbits, and if $Ω$ is the (maximal) boundary of $\cB$, then $C(\Om)\rtimes \G$ is a p.i.s.u.n. $C^*$-algebra. This algebra has a structure theory analogous to that of a simple Cuntz-Krieger algebra and is the motivation for a theory of higher rank Cuntz-Krieger algebras, which has been developed by T. Steger and G. Robertson. The K-theory of these algebras can be computed explicitly in the rank two case. For the rank two examples of t…

Topological conjugacy of topological Markov shifts and Cuntz-Krieger algebras

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…

关于Cuntz--Krieger代数的强扩张群

On strong extension groups of Cuntz--Krieger algebras

本文研究Cuntz--Krieger代数的强扩张群，给出计算公式，并证明带Toeplitz扩张位置的弱扩张群是该代数同构类的完全不变量。

A generalized Cuntz-Krieger uniqueness theorem for higher rank graphs

We present a uniqueness theorem for k-graph C*-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k-graph C*-algebra, it is sufficient that the representation be injective on a distinguished abelian C*-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C*-algebra, each of which is the unique extension of a state on the dist…

Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz--Krieger algebras

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.

K-theory for Cuntz-Krieger algebras arising from real quadratic maps

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 $f_μ$.

State splitting, strong shift equivalence and stable isomorphism of Cuntz-Krieger algebras

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…

Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz--Krieger algebras

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.

On a family of $C^*$-subalgebras of Cuntz-Krieger algebras

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.

Quantum Cuntz-Krieger algebras

Motivated by the theory of Cuntz-Krieger algebras we define and study $ C^\ast $-algebras associated to directed quantum graphs. For classical graphs the $ C^\ast $-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to …

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.

The $C^*$-algebras of finitely aligned higher-rank graphs

We generalise the theory of Cuntz-Krieger families and graph algebras to the class of finitely aligned $k$-graphs. This class contains in particular all row-finite $k$-graphs. The Cuntz-Krieger relations for non-row-finite $k$-graphs look significantly different from the usual ones, and this substantially complicates the analysis of the graph algebra. We prove a gauge-invariant uniqueness theorem and a Cuntz-Krieger uniqueness theorem for the $C^*$-algebras of finitely aligned $k$-graphs.

Full Cuntz-Krieger dilations via non-commutative boundaries

We apply Arveson's non-commutative boundary theory to dilate every Toeplitz-Cuntz-Krieger family of a directed graph $G$ to a full Cuntz-Krieger family for $G$. We do this by describing all representations of the Toeplitz algebra $\mathcal{T}(G)$ that have unique extension when restricted to the tensor algebra $\mathcal{T}_+(G)$. This yields an alternative proof to a result of Katsoulis and Kribs that the $C^*$-envelope of $\mathcal T_+(G)$ is the Cuntz-Krieger algebra $\mathcal O(G)$. We then generalize our dilation results further, to the context of colored directed graphs, by investigating …

C$^{*}$-bialgebra defined by the direct sum of Cuntz-Krieger algebras

Let ${\sf CK}_{*}$ denote the C$^{*}$-algebra defined by the direct sum of all Cuntz-Krieger algebras. We introduce a comultiplication $Δ_ϕ$ and a counit $ε$ on ${\sf CK}_{*}$ such that $Δ_ϕ$ is a nondegenerate $*$-homomorphism from ${\sf CK}_{*}$ to ${\sf CK}_{*}\otimes {\sf CK}_{*}$ and $ε$ is a $*$-homomorphism from ${\sf CK}_{*}$ to ${\bf C}$. From this, ${\sf CK}_{*}$ is a counital non-commutative non-cocommutative C$^{*}$-bialgebra. Furthermore, C$^{*}$-bialgebra automorphisms, a tensor product of representations and C$^{*}$-subbialgebras of ${\sf CK}_{*}$ are investigated.

互反Cuntz--Krieger代数

Reciprocal Cuntz--Krieger algebras

本文通过K-理论对偶构造Kirchberg代数的互反对偶代数，具体实现Cuntz--Krieger代数的互反代数，并证明其自同构群基本群同构且保持规范作用。

Reduction of filtered K-theory and a characterization of Cuntz-Krieger algebras

We show that filtered K-theory is equivalent to a substantially smaller invariant for all real-rank-zero C*-algebras with certain primitive ideal spaces -- including the infinitely many so-called accordion spaces for which filtered K-theory is known to be a complete invariant. As a consequence, we give a characterization of purely infinite Cuntz-Krieger algebras whose primitive ideal space is an accordion space.

Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for infinite matrices

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.

On freely generated semigraph $C^*$-algebras

For special universal $C^*$-algebras associated to $k$-semigraphs we present the universal representations of these algebras, prove a Cuntz--Krieger uniqueness theorem, and compute the $K$-theory. These $C^*$-algebras seem to be the most universal Cuntz--Krieger like algebras naturally associated to $k$-semigraphs. For instance, the Toeplitz Cuntz algebra is a proper quotient of such an algebra.

K-theoretic duality for extensions of Cuntz-Krieger algebras

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 pa…

Hyperrigid subsets of Cuntz-Krieger algebras and the property of rigidity at zero

A subset $\mathcal{G}$ generating a $C^*$-algebra $A$ is said to be hyperrigid if for every faithful nondegenerate $*$-representation $A\subseteq B(H)$ and a sequence $ϕ_n:B(H) \to B(H)$ of unital completely positive maps, we have that \[ \lim_{n\to\infty}ϕ_n(g)= g~~\text{for all } g\in \mathcal{G} ~~ \implies ~~ \lim_{n\to\infty}ϕ_n(a)= a~~\text{for all } a\in A \] where all convergence are in norm. In this paper, we show that for the Cuntz-Krieger algebra $\mathcal{O}(G)$ associated to a row-finite directed graph $G$ with no isolated vertices, the set of partial isometries $\mathcal{E}=\{S_e…

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 $C(X_A)$.

Higher-rank graphs and their C*-algebras

We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz-Krieger algebras. We describe a variant of the Cuntz-Krieger relations which applies to graphs with sources, and describe a local convexity condition which characterises the higher-rank graphs that admit a nontrivial Cuntz-Krieger family. We then prove versions of the uniqueness theorems and classifications of ideals for the C*-algebras generated by Cuntz-Krieger families.