Aidan Sims
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
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.
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.
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…
Let E be an arbitrary graph and K be any field. For every non-graded ideal I of the Leavitt path algebra L_{K}(E), we give an explicit description of the generators of I. Using this, we show that every finitely generated ideal of L_{K}(E) must be principal. In particular, if E is a finite graph, then every ideal of L_{K}(E) must be principal ideal.
The class of AD algebras of real rank zero is classified by an exact sequence of K-groups with coefficients, equipped with certain order structures. Such a sequence is always split, and one may ask why, then, the middle group is relevant for classification. The answer is that the splitting map can not always be chosen to respect the order structures involved. This may be rephrased in terms of the ideals of the C*-algebras in question. We prove that when the C*-algebra has only finitely many ideals, a splitting map respecting these always exists. Hence AD algebras of real rank zero with finitel…
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。
Consider the Deaconu-Renault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact Hausdorff space. We catalogue the primitive ideals of the associated groupoid C*-algebra. For a special class of actions we describe the Jacobson topology.
We consider the problem of identifying exactly which AF-algebras are isomorphic to a graph C*-algebra. We prove that any separable, unital, Type I C*-algebra with finitely many ideals is isomorphic to a graph C*-algebra. This result allows us to prove that a unital AF-algebra is isomorphic to a graph C*-algebra if and only if it is a Type I C*-algebra with finitely many ideals. We also consider nonunital AF-algebras that have a largest ideal with the property that the quotient by this ideal is the only unital quotient of the AF-algebra. We show that such an AF-algebra is isomorphic to a graph …
Let A be a simple, sigma-unital, non-unital C*-algebra, with metrizable tracial simplex T(A), which is projection-surjective and injective and has strict comparison of positive elements by traces. Then the following are equivalent: (i) A has quasicontinuous scale; (ii) The multiplier algebra M(A) has strict comparison of positive elements by traces; (iii) The coronal algebra M(A)/A is purely infinite; (iii') The quotient M(A)/Imin is purely infinite; (iv) M(A) has finitely many ideals; (v) Imin=Ifin. If furthermore algebra of n by n matrices of A is projection-surjective and injective for ever…
We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$: (Q1) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (Q2) Is every finitely-generated, maximal left ideal of $\mathscr{B}(E)$ necessarily of the form \{T\in\mathscr{B}(E): Tx = 0\} (*) for some non-zero $x\in E$? Since the two-sided ideal $\mathscr{F}(E)$ of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to the se…
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 …
Orbit equivalence of higher-rank graphs
本文研究了高秩图的连续轨道等价、最终单侧共轭和双侧共轭，证明了保持周期性的连续轨道等价等价于边界路径群胚同构，并用$C^*$-代数和Kumjian-Pask代数刻画了这些等价关系。
On the primitive ideal space of radial representations of free groups
本文研究自由群径向表示的原始理想空间，揭示其在C*-代数中的拓扑结构。
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。
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。