Takeshi Katsura, Paul S. Muhly, Aidan Sims, Mark Tomforde
Given an ultragraph in the sense of Tomforde, we construct a topological quiver in the sense of Muhly and Tomforde in such a way that the universal C*-algebras associated to the two objects coincide. We apply results of Muhly and Tomforde for topological quiver algebras and of Katsura for topological graph C*-algebras to study the K-theory and gauge-invariant ideal structure of ultragraph C*-algebras.
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We prove two isomorphism-invariance theorems for groupoids associated with ultragraphs. These theorems characterize ultragraphs for which the topological full group of an associated groupoid is an isomorphism invariant. These results extend those of graph groupoids to ultragraph groupoids while providing another concrete example where the topological full group of a groupoid is a complete isomorphism invariant.
Given a partial action $α$ of a groupoid $G$ on a ring $R$, we study the associated partial skew groupoid ring $R \rtimes_α G$, which carries a natural $G$-grading. We show that there is a one-to-one correspondence between the $G$-invariant ideals of $R$ and the graded ideals of the $G$-graded ring $R \rtimes_αG.$ We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of $R \rtimes_αG.$ We show that every ideal of $R \rtimes_αG$ is graded if, and only if, $α$ has the residual intersection property. Furthermore, if $α$ is induced by a topological …
An ultragraph gives rise to a labelled graph with some particular properties. In this paper we describe the algebras associated to such labelled graphs as groupoid algebras. More precisely, we show that the known groupoid algebra realization of ultragraph C*-algebras is only valid for ultragraphs for which the range of each edge is finite, and we extend this realization to any ultragraph (including ultragraphs with sinks). Using our machinery, we characterize the shift space associated to an ultragraph as the tight spectrum of the inverse semigroup associated to the ultragraph (viewed as a lab…
On morphisms of topological quivers
本文引入拓扑箭图的正则态射，证明其构成子范畴，并证明该正则条件使拓扑箭图C*-代数构造成为反变函子。
Coactions and Skew Products for Topological Quivers
本文研究拓扑quiver的余作用与斜积，构造斜积拓扑quiver并给出识别条件，证明斜积C*-代数与原quiver的余作用交叉积同构，且对偶作用的约化交叉积与原C*-代数Morita等价。
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 …
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…
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…
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…
Principal Actions on Topological Quivers and Associated Operator Dynamics
该研究针对容许局部紧群自由真作用的拓扑箭图，给出其拓扑分类，构建了轨道箭图C*-代数与原箭图C*-代数不动点代数的同构，并考察了相关实例与Morita等价。
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 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…
A Markov operator $P$ acting on $C(X)$, where $X$ is compact, gives rise to a natural topological quiver. We use the theory of such quivers to attach a $C^{*}$-algebra to $P$ in a fashion that reflects some of the probabilistic properties of $P$.
Equivalence of categories of bivariant K-theory for C*-algebras over topological spaces via reflection functors
本文通过引入反射函子，证明了拓扑空间上C*-代数的双变量K-理论相关范畴的等价性，还将其应用于滤过K-理论的通用系数定理证明。
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