Jacob Shotwell
It is shown that no local periodicity is equivalent to the aperiodicity condition for arbitrary finitely-aligned k-graphs. This allows us to conclude that C*(Λ) is simple if and only if Λis cofinal and has no local periodicity.
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We define the relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs. We prove versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for relative Cuntz-Krieger algebras.
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.
We produce a complete descrption of the lattice of gauge-invariant ideals in $C^*(Λ)$ for a finitely aligned $k$-graph $Λ$. We provide a condition on $Λ$ under which every ideal is gauge-invariant. We give conditions on $Λ$ under which $C^*(Λ)$ satisfies the hypotheses of the Kirchberg-Phillips classification theorem.
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.
We introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs Λ, and prove that C*(Λ) is simple if and only if Λis aperiodic and cofinal. The main advantage of our versions of aperiodicity and cofinality over existing ones is that ours are stated in terms of finite paths. To prove our main result, we first characterise each of aperiodicity and cofinality of Λin terms of the ideal structure of C*(Λ). In an appendix we show how our new cofinality condition simplifies in a number of special cases which have been treated previously in the literature; even in …
We consider Toeplitz and Cuntz-Krieger $C^*$-algebras associated with finitely aligned left cancellative small categories. We pay special attention to the case where such a category arises as the Zappa-Szép product of a category and a group linked by a one-cocycle. As our main application, we obtain a new approach to Exel-Pardo algebras in the case of row-finite graphs. We also present some other ways of constructing $C^*$-algebras from left cancellative small categories and discuss their relationship.
We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish versions of the usual uniqueness theorems and the classification of gauge-invariant ideals. We show that all twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs are nuclear and satisfy the UCT, and that for twists that lift to real-valued cocycles, the K-theory of a twisted relative Cuntz-Krieger algebra is independent of the twist. In the final section, we identif…
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…
An inverse semigroup approach to self-similar k-graph $C^*$-algebras and simplicity
本文将Li-Yang的自相似k-图及其C*-代数推广到任意有限对齐k-图，并借助逆半群模型分析其紧群胚与C*-代数结构，为研究单纯性提供了新工具。
Groupoids of finitely aligned higher-rank graphs via filters and graph morphisms
本文推广图态射方法至有限对齐P-图，证明滤子与图态射方法产生同构的路径和边界路径群胚，并综述多种同构表示。
A local treatment of finite alignment and path groupoids of nonfinitely aligned higher-rank graphs
本文提出了一种对非有限对齐高阶图的局部处理方法，定义了新的局部紧致路径和边界路径空间，并扩展了半群作用以构建 ample Hausdorff 路径和边界路径群oid。
Let (G,S) be a finitely generated Coxeter group, such that the Coxeter system is indecomposable and the canonical bilinear form is indefinite but non-degenerate. We show that the reduced C-*-algebra of G is simple with unique normalised trace. For an arbitrary finitely generated Coxeter group we prove the validity of a Haagerup inequality: There exist constants C>0 and a natural number L such that for a function f in l^2(G) supported on elements of length n with respect to the generating set S: || f * h || <= C(n+1)^{3/2 L} || f || || h ||, forall h in l^2(G).
The main result here is that a simple separable C*-algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of [Toms, "K-theoretic rigidity and slow dimension growth"; Winter, "Nuclear dimension and Z-stability of pure C*-algebras"] to the nonunital setting. As a consequence, finite nuclear dimension implies Z-stability even in the case of a separable C*-algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of…
We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in N. We focus on semigroups P arising as part of a quasi-lattice ordered group (G,P) in the sense of Nica, and on P-graphs which are finitely aligned in the sense of Raeburn and Sims. We show that each finitely aligned P-graph admits a C*-algebra C*_{min}(Lambda) which is co-universal for partial-isometric representations of Lambda which admit a coaction of G compatible with the P-valued length function. We also characterise when a homomorphism induced by the …
We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $Λ$. We identify the boundary-path space $\partialΛ$ as the spectrum of a commutative $C^*$-subalgebra $D_Λ$ of $C^*(Λ)$. Then, using a construction similar to that of Farthing, we construct a finitely aligned $k$-graph $\wtΛ$ with no sources in which $Λ$ is embedded, and show that $\partialΛ$ is homeomorphic to a subset of $\partial\wtΛ$ . We show that when $Λ$ is row-finite, we can identify $C^*(Λ)$ with a full corner of $C^*(\wtΛ)$, and deduce that $D_Λ$ is isomorphic to a corner of $D_{\wtΛ}…
We use the boundary-path space of a finitely-aligned k-graph Λto construct a compactly-aligned product system X, and we show that the graph algebra C^*(Λ) is isomorphic to the Cuntz-Nica-Pimsner algebra NO(X). In this setting, we introduce the notion of a crossed product by a semigroup of partial endomorphisms and partially-defined transfer operators by defining it to be NO(X). We then compare this crossed product with other definitions in the literature.
Nekrashevych algebras of self-similar group actions are natural generalizations of the classical Leavitt algebras. They are discrete analogues of the corresponding Nekrashevych $C^\ast$-algebras. In particular, Nekrashevych, Clark, Exel, Pardo, Sims and Starling have studied the question of simplicity of Nekrashevych algebras, in part, because non-simplicity of the complex algebra implies non-simplicity of the $C^\ast$-algebra. In this paper we give necessary and sufficient conditions for the Nekrashevych algebra of a contracting group to be simple. Nekrashevych algebras of contracting groups …
We provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the Cuntz-Krieger algebra.
Self-similar groupoid actions on k-graphs, and invariance of K-theory for cocycle homotopies
本文研究了有限对齐左消去范畴的包含与扭曲C*-代数包含的关系，并证明了可数离散顺从群胚在无源行有限k-图上的联合忠实自相似作用，在同伦余循环下，其扭曲C*-代数的K-理论不变。
Finite-dimensional approximations for C*-algebras of product systems
研究积系统生成的C*-代数的精确性与核性，证明其等价于核或系数代数的相应性质，并指出紧对齐假设的必要性。
For a finitely aligned k-graph $Λ$ with X a set of vertices in $Λ$ we define a universal C*-algebra called $C^*(Λ,X)$ generated by partial isometries. We show that $C^*(Λ,X)$ is isomorphic to the corner $P_XC^*(Λ)P_X$, where $P_X$ is the sum of vertex projections in X. We then prove a version of the Gauge Invariant Uniqueness Theorem for $C^*(Λ,X)$, and then use the theorem to prove various results involving fullness, simplicity and Morita equivalence as well as new results involving symbolic dynamics.
We define branching systems for finitely aligned higher-rank graphs. From these we construct concrete representations of higher-rank graph C*-algebras on Hilbert spaces. We prove a generalized Cuntz-Krieger uniqueness theorem for periodic single-vertex 2-graphs. We use this result to give a sufficient condition under which representations of periodic single-vertex 2-graph C*-algebras arising from branching systems are faithful.
Let (G,P) be a quasi-lattice ordered group and let X be a compactly aligned product system over P of Hilbert bimodules. Under mild hypotheses we associate to X a C*-algebra which we call the Cuntz-Nica-Pimsner algebra of X. Our construction generalises a number of others: a sub-class of Fowler's Cuntz-Pimsner algebras for product systems of Hilbert bimodules; Katsura's formulation of Cuntz-Pimsner algebras of Hilbert bimodules; the C*-algebras of finitely aligned higher-rank graphs; and Crisp and Laca's boundary quotients of Toeplitz algebras. We show that for a large class of product systems …
We construct a representation of each finitely aligned aperiodic k-graph Λ on the Hilbert space H^{ap} with basis indexed by aperiodic boundary paths in Λ. We show that the canonical expectation on B(H^{ap}) restricts to an expectation of the image of this representation onto the subalgebra spanned by the final projections of the generating partial isometries. We then show that every quotient of the Toeplitz algebra of the k-graph admits an expectation compatible with this one. Using this, we prove that the image of our representation, which is canonically isomorphic to the Cuntz-Krieger algeb…
Growth conditions for topological freeness
研究有限生成群在紧空间上的极小作用，通过增长/衰减函数给出拓扑自由性的充分条件，并应用于Furstenberg边界作用以得到C*-单性。
Equivariant Nica-Pimsner quotients associated with strong compactly aligned product systems
该研究通过满足不变性、偏序性和极大性的2^d元组参数化Z_+^d上强紧对齐乘积系统的Toeplitz-Nica-Pimsner代数的规范不变理想，给出极大性刻画与格同构条件，还提供广义规范不变唯一性定理并应用于多类系统，与已知结果兼容。
Given a finitely aligned $k$-graph $Λ$, we let $Λ^i$ denote the $(k-1)$-graph formed by removing all edges of degree $e_i$ from $Λ$. We show that the Toeplitz-Cuntz-Krieger algebra of $Λ$, denoted by $\mathcal{T}C^*(Λ)$, may be realised as the Toeplitz algebra of a Hilbert $\mathcal{T}C^*(Λ^i)$-bimodule. When $Λ$ is locally-convex, we show that the Cuntz-Krieger algebra of $Λ$, which we denote by $C^*(Λ)$, may be realised as the Cuntz-Pimsner algebra of a Hilbert $C^*(Λ^i)$-bimodule. Consequently, $\mathcal{T}C^*(Λ)$ and $C^*(Λ)$ may be viewed as iterated Toeplitz and iterated Cuntz-Pimsner al…
Non-Hausdorff etale groupoids and C*-algebras of left cancellative monoids
该文研究局部紧平展广群单位空间稠子集表示确定约化范数的问题，给出充分条件与无挠迷向群情形的完整解答，提出左可消幺半群关联广群的C*-正则性概念并构造反例。
Uniqueness theorems for combinatorial C*-algebras
本文基于已知的群oid模型和Exel的逆半群紧表示理论，证明了多种组合C*-代数的唯一性定理，并改进了边界商C*-代物的唯一性定理，同时将Brown等人的高阶图唯一性定理推广到有限对齐情况。
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