Taro Sogabe
For a continuous field of the Cuntz algebra over a finite CW complex, we introduce a topological invariant, which is an element in Dadarlat-Pennig's generalized cohomology group, and prove that the invariant is trivial if and only if the field comes from a vector bundle via Pimsner's construction.
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Topological full groups arising from Cuntz and Cuntz-Toeplitz algebras and their crossed products
论文研究 Cuntz 与 Cuntz-Toeplitz 代数及其 Cartan 交叉积所诱导的拓扑满群，分析其正规子群和交换化，并完全刻画相关约化交叉积在规范作用下的 KMS 态。
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.
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.
A study of noncommutative topological entropy of gauge invariant endomorphisms of Cuntz algebras began in our earlier work with Joachim Zacharias is continued and extended to endomorphisms which are not necessarily of permutation type. In particular it is shown that if H is an N-dimensional Hilbert space, V is an irreducible multiplicative unitary on the tensor product of H with itself and F is the tensor flip, then the Voiculescu entropy of the Longo's canonical endomorphism associated with the unitary VF is equal to log N.
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.
Let $A$ be an $N\times N$ irreducible matrix with entries in $\{0,1\}$. We present an easy way to find an $(N+3)\times (N+3)$ irreducible matrix $\bar{A}$ with entries in $\{0,1\}$ such that their Cuntz--Krieger algebras ${\mathcal{O}}_A$ and ${\mathcal{O}}_{\bar{A}}$ are isomorphic and $ \det(1 -A) = - \det(1-\bar{A}). $ As a consequence, we know that two Cuntz--Krieger algebras ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$ are isomorphic if and only if the one-sided topological Markov shift $(X_A, σ_A)$ is continuously orbit equivalent to $(X_B, σ_B)$ or $(X_{\bar{B}}, σ_{\bar{B}}).$
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 compute the sheaf cohomology of a groupoid built from a local homeomorphism of a locally compact space $X$. In particular, we identify the twists over this groupoid, and its Brauer group. Our calculations refine those made by Kumjian, Muhly, Renault and Williams in the case $X$ is the path space of a graph, and the local homeomorphism is the shift. We also show how the C*-algebra of a twist may be identified with the Cuntz-Pimsner algebra constructed from a certain C*-correspondence.
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.
This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C_0(X,A), where A is a unital, simple, Z-stable ASH algebra. The computations describe the Cuntz semigroup in terms of Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(T,A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of well-behaved non-simple C*-algebras.
We investigate $C^*$-algebras associated with row-finite topological higher-rank graphs with no source, which are based on product system $C^*$-algebras. We prove the Cuntz-Krieger uniqueness theorem, and provide the condition of simplicity and purely infiniteness of our algebras. We give examples of non-discrete topological higher-rank graphs whose $C^*$-algebras contain Cuntz's $ax+b$-semigroup $C^*$-algebra over $\mathbb{N}$.
We investigate an invariant for continuous fields of the Cuntz algebra $\mathcal{O}_{n+1}$ introduced in our previous work, and find a way to obtain a continuous field of $\mathbb{M}_n(\mathcal{O}_\infty)$ from that of $\mathcal{O}_{n+1}$ using the construction of the invariant. By Brown's representability theorem, this gives a bijection from the set of the isomorphism classes of continuous fields of $\mathcal{O}_{n+1}$ to those of $\mathbb{M}_n(\mathcal{O}_\infty)$. As a consequence, we obtain a new proof for M. Dadarlat's classification result of continuous fields of $\mathcal{O}_{n+1}$ aris…
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 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.
Noncommutative topological entropy estimates are obtained for polynomial gauge invariant endomorphisms of Cuntz algebras, generalising known results for the canonical shift endomorphisms. Exact values for the entropy are computed for a class of permutative endomorphisms related to branching function systems introduced and studied by Bratteli, Jorgensen and Kawamura.
We show that the passage from a $C^\ast$-correspondence to its Cuntz-Pimsner $C^\ast$-algebra gives a functor on a category of $C^\ast$-correspondences with appropriately defined morphisms. Applications involving topological graph $C^\ast$-algebras are discussed, and an application to crossed-product correspondences is presented in detail.
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 $C(X_A)$.
Groupoid models for relative Cuntz-Pimsner algebras of groupoid correspondences
本文研究了群oid对应关系诱导的C*-对应关系的相对Cuntz-Pimsner代数，并展示了其作为特定群oid的C*-代数的结构，统一了拓扑图和自相似群的C*-代数的构造方法。
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.
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).
Let H be a full Hilbert bimodule over a C*-algebra A. We show that the Cuntz-Pimsner C*-algebra associated to H is exact if and only if A is exact. Using this result, we give alternative proofs for exactness of reduced amalgamated free products of exact C*-algebras. In the case that A is a finite dimensional C*-algebra, we also show that the Brown-Voiculescu topological entropy of Bogljubov automorphisms of the Cuntz-Pimsner algebra associated to an A,A Hilbert bimodule is zero.
We prove stability theorems in the Cuntz semigroup of a commutative C*-algebra which are analogues of classical stability theorems for topological vector bundles over compact Hausdorff spaces. Several applications to simple unital AH algebras of slow dimension growth are then given: such algebras have strict comparison of positive elements; their Cuntz semigroups are recovered functorially from the Elliott invariant; the lower-semicontinuous dimension functions are dense in the space of all dimension functions, and the latter is a Choquet simplex.
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 revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of 1-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: We show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.
We prove a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type. To this end, we introduce Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product system of 'regular' C*-correspondences. Under a certain aperiodicity condition on the latter, we obtain the uniqueness theorem and a simplicity criterion for the algebras in question. These results generalize the corresponding ones for crossed products by discrete groups, due to Archbold and Spielberg, and for Exel's crossed products, due to Exel and Vershik…
Contractible Cuntz classes in $Z$-stable C$^*$-algebras
本文证明单的$Z$-稳定C$^*$-代数中固定非紧Cuntz类的正元素集合是可缩的，并结合紧情形早期结果，完整计算了这些代数Cuntz类的同伦群。
In this paper we describe the Cuntz semigroup of continuous fields of C$^*$-algebras over one dimensional spaces whose fibers have stable rank one and trivial $K_1$ for each closed, two-sided ideal. This is done in terms of the semigroup of global sections on a certain topological space built out of the Cuntz semigroups of the fibers of the continuous field. When the fibers have furthermore real rank zero, and taking into account the action of the space, our description yields that the Cuntz semigroup is a classifying invariant if and only if so is the sheaf induced by the Murray-von Neumann s…
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 determine the group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra $O_{n+1}$ for finite n in terms of K-theory. We show that there is an example of a space for which the homotopy set is a non-commutative group, and hence the classifying space of the automorphism group of the Cuntz algebra for finite n is not an H-space. We also make an improvement of Dadarlat's classification of continuous fields of the Cuntz algebras in terms of vector bundles.
We study topological aspects of the category of abstract Cuntz semigroups, termed Cu. We provide a suitable setting in which we are able to uniformly control how to approach an element of a Cu-semigroup by a rapidly increasing sequence. This approximation induces a semimetric on the set of Cu-morphisms, generalizing Cu-metrics that had been constructed in the past for some particular cases. Further, we develop an approximate intertwining theory for the category Cu. Finally, we give several applications such as the classification of unitary elements of any unital AF-algebra by means of the func…
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…
In this paper, we show that for unital, separable $C^*$-algebras of stable rank one and real rank zero, the unitary Cuntz semigroup functor and the functor ${\rm K}_*$ are naturallly equivalent. Then we introduce a refinement of the unitary Cuntz semigroup, say the total Cuntz semigroup, which is a new invariant for separable $C^*$-algebras of stable rank one, is a well-defined continuous functor from the category of $C^*$-algebras of stable rank one to the category ${\rm\underline{ Cu}}$. We prove that this new functor and the functor ${\rm \underline{K}}$ are naturallly equivalent for unital…
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。
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.
Using free probability constructions involving Cuntz-Pimsner C*-algebras we show that the topological entropy of the free product of two automorphisms is equal to the maximum of the individual entropies. As applications we show that general free shifts have entropy zero. We show that any nuclear C*-dynamical system admits an entropy preserving covariant embedding into the Cuntz algebra on infinitely many generators. It follows that any simple nuclear purely infinite C*-algebra admits an automorphism with any given value of entropy. As a final application we show that if two automorphisms satis…
We study the Pimsner algebra associated with the module of continuous sections of a Hilbert bundle, and prove that it is a continuous bundle of Cuntz algebras. We discuss the role of such Pimsner algebras w.r.t. the notion of inner endomorphism. Furthermore, we study bundles of Cuntz algebras carrying a global circle action, and assign to them a class in the representable KK-group of the zero-grade bundle. We compute such class for the Pimsner algebra of a vector bundle.
Topological quivers generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver $Q$ is a $C^*$-correspondence, and in turn, a Cuntz-Pimsner algebra $C^*(Q).$ Given $Γ$ a locally compact group and $α$ and $β$ endomorphisms on $Γ,$ one may construct a topological quiver $Q_{α,β}(Γ)$ with vertex set $Γ,$ and edge set $Ω_{α,β}(Γ)= \{(x,y)\inΓ\timesΓ\| α(y)=β(x)\}.$ In \cite{Mc1}, the author examined the Cuntz-Pimsner algebra $\cal{O}_{α,β}(Γ):=C^*(Q_{α,β}(Γ))$ and found generators (and …
Let $A$ be a separable simple exact ${\cal Z}$-stable $C^*$-algebra. We show that the unitay group of ${\tilde A}$ has the cancellation property. If $A$ has continuous scale, the Cuntz semigroup of $\tilde A$ has the strict comparison property and a weak cancellation property. Let $C$ be a 1-dimensional non-commutative CW complex with $K_1(C)=\{0\}.$ Suppose that $λ: {\rm Cu}^\sim(C)\to {\rm Cu}^\sim(A)$ is a morphism in Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms $ϕ_n: C\to A$ such that $\lim_{n\to\infty}{\rm Cu}^\sim(ϕ_n)=λ.$ This result leads t…
Topological quivers generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver $Q$ is a $C^*$-correspondence, and in turn, a Cuntz-Pimsner algebra $C^*(Q).$ Given $Γ$ a locally compact group and $α$ and $β$ endomorphisms on $Γ,$ one may construct a topological quiver $Q_{α,β}(Γ)$ with vertex set $Γ,$ and edge set $Ω_{α,β}(Γ)= \{(x,y)\inΓ\timesΓ\st α(y)=β(x)\}.$ In \cite{Mc1}, the author examined the Cuntz-Pimsner algebra $\cO_{α,β}(Γ):=C^*(Q_{α,β}(Γ))$ and found generators (and the…
Topological quivers generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver $Q$ is a $C^*$-correspondence, and in turn, a Cuntz-Pimsner algebra $C^*(Q).$ Given $Γ$ a locally compact group and $α$ and $β$ endomorphisms on $Γ,$ one may construct a topological quiver $Q_{α,β}(Γ)$ with vertex set $Γ,$ and edge set $Ω_{α,β}(Γ)= \{(x,y)\inΓ\timesΓ\st α(y)=β(x)\}.$ In this paper, the author examines the Cuntz-Pimsner algebra $\cO_{α,β}(Γ):=C^*(Q_{α,β}(Γ)).$ The investigative topics inc…
We interpret the Cuntz-Pimsner covariance condition as a nondegeneracy condition for representations of product systems. We show that Cuntz-Pimsner algebras over Ore monoids are constructed through inductive limits and section algebras of Fell bundles over groups. We construct a groupoid model for the Cuntz-Pimsner algebra coming from an action of an Ore monoid on a space by topological correspondences. We characterise when this groupoid is effective or locally contracting and describe its invariant subsets and invariant measures.
We introduce the notion of a topological higher-rank graph, a unified generalization of the higher-rank graph and the topological graph. Using groupoid techniques, we define the Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs, and show that the $C^*$-algebras defined are coherent with the existing theory.
We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra ${\cal O}_2.$ We further prove that if $A$ is a simple separable unital nuclear C*-algebra, then ${\cal O}_2 \otimes A \cong {\cal O}_2,$ and if, in addition, $A$ is purely infinite, then ${\cal O}_{\infty} \otimes A \cong A.$ The embedding of exact C*-algebras in $\OA{2}$ is continuous in the following sense. If $A$ is a continuous field of C*-algebras over a compact manifold or finite CW complex $X$ with fiber $A (x)$ over $x \in X,$ such that the algebra of continuous sections of $A$ is separable…
Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver Q is a C*-correspondence, and from this correspondence one may construct a Cuntz-Pimsner algebra C*(Q). In this paper we develop the general theory of topological quiver C*-algebras and show how certain C*-algebras found in the literature may be viewed from this general perspective. In particular, we show that C*-algebras of topological quivers generalize the well-studied class of graph C*-algebras and in analogy with t…
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。
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 provide groupoid models for Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs. Extending the groupoid models used in the theory of graph algebras and topological dynamical systems to our setting, we prove results on essential freeness and amenability of the groupoids which capture the existing theory, and extend results involving group crossed products of graph algebras.
We introduce a new invariant for C*-algebras of stable rank one that merges the Cuntz semigroup information together with the K$_1$-group information. This semigroup, termed the Cu$_1$-semigroup, is constructed as equivalence classes of pairs consisting of a positive element in the stabilization of the given C*-algebra together with a unitary element of the unitization of the hereditary subalgebra generated by the given positive element. We show that the Cu$_1$-semigroup is a well-defined continuous functor from the category of C*-algebras of stable rank one to a suitable codomain category tha…
Let $A$ be a simple, separable C$^*$-algebra of stable rank one. We prove that the Cuntz semigroup of $\CC(\T,A)$ is determined by its Murray-von Neumann semigroup of projections and a certain semigroup of lower semicontinuous functions (with values in the Cuntz semigroup of $A$). This result has two consequences. First, specializing to the case that $A$ is simple, finite, separable and $\mathcal Z$-stable, this yields a description of the Cuntz semigroup of $\CC(\T,A)$ in terms of the Elliott invariant of $A$. Second, suitably interpreted, it shows that the Elliott functor and the functor def…
KMS states of quasi-free dynamics on Pimsner algebras
本研究刻画了右希尔伯特C*-双模上连续单参数酉等距群诱导的拟自由动力学的KMS态，推广了此前Cuntz代数等的相关结果，还给出了非酉等距情形下基于KMS权的一般刻画。
We use product systems of $C^*$-correspondences to introduce twisted $C^*$-algebras of topological higher-rank graphs. We define the notion of a continuous $\mathbb{T}$-valued $2$-cocycle on a topological higher-rank graph, and present examples of such cocycles on large classes of topological higher-rank graphs. To every proper, source-free topological higher-rank graph $Λ$, and continuous $\mathbb{T}$-valued $2$-cocycle $c$ on $Λ$, we associate a product system $X$ of $C_0(Λ^0)$-correspondences built from finite paths in $Λ$. We define the twisted Cuntz--Krieger algebra $C^*(Λ,c)$ to be the C…
We define the notion of a twisted topological graph algebra associated to a topological graph and a $1$-cocycle on its edge set. We prove a stronger version of a Vasselli's result. We expand Katsura's results to study twisted topological graph algebras. We prove a version of the Cuntz-Krieger uniqueness theorem, describe the gauge-invariant ideal structure. We find that a twisted topological graph algebra is simple if and only if the corresponding untwisted one is simple.
We study conditions that ensure uniqueness theorems of Cuntz-Krieger type for relative Cuntz-Pimsner algebras $\mathcal{O}(J,X)$ associated to a $C^*$-correspondence $X$ over a $C^*$-algebra $A$. We give general sufficient conditions phrased in terms of a multivalued map $\widehat{X}$ acting on the spectrum $\widehat{A}$ of $A$. When $X(J)$ is of Type I we construct a directed graph dual to $X$ and prove a uniqueness theorem using this graph. When $X(J)$ is liminal, we show that topological freeness of this graph is equivalent to the uniqueness property for $\mathcal{O}(J,X)$, as well as to an…
Quantum isometry groups of log-Laplacians on Cuntz--Krieger algebras
研究Cuntz-Krieger代数的量子等距群，结合拓扑马尔可夫链的Ahlfors正则结构，提出新的紧量子群家族，展现图的量子自同构群与易量子群的混合特性，并构造了Cantor空间上的量子 ergodic 行动。
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.
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…
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。
Let $A$ be a simple C*-algebra of stable rank one and let $p$ and $q$ be two $σ$-compact open projections. It is proved that there is a continuous path of unitaries in ${\tilde A}$ which connects open sub-projections of $p$ which is compactly contained in $p$ to those in $q.$ It is also shown that every Hilbert module is projective in the category whose morphisms are bounded module maps with adjoints. A discussion of projective Hilbert modules (whose morphisms are bounded module maps) is also given.
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…
Let $(X, Γ)$ be a free minimal dynamical system, where $X$ is a compact separable Hausdorff space and $Γ$ is a discrete amenable group. It is shown that, if $(X, Γ)$ has a version of Rokhlin property (uniform Rokhlin property) and if $\mathrm{C}(X)\rtimesΓ$ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes Γ$ is at most half of the mean topological dimension of $(X, Γ)$. These two conditions are shown to be satisfied if $Γ= \mathbb Z$ or if $(X, Γ)$ is an extension of a free Cantor system and $Γ$ has subexponential growth. …
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