Remarks on some fundamental results about higher-rank graphs and their C*-algebras
Remarks on some fundamental results about higher-rank graphs and their C*-algebras
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关于高阶图及其 C* 代数的一些基本结果的评论
DOI：
10.1017/s0013091512000338
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发表时间：
2011
影响因子：
0.7
通讯作者：
Samuel B. G. Webster
中科院分区：
文献类型：
--
作者：
R. Hazlewood;I. Raeburn;A. Sims;Samuel B. G. Webster
Fowler和Sims的抽象结果表明，每个k-图完全由它的k-色骨架和交换方块的集合决定。本文给出了与给定骨架和平方集相关的k-图的一个显式刻画，并证明了两个k-图同构的充要条件是它们的骨架同构，且保持交换平方。利用这一点，我们直接证明了每个k-图Λ通过交换平方所确定的等价关系与其骨架的路范畴的商同构，并证明了当k-图是行有限的无源图时，这推广到无限路空间的同态。最后，我们用一个简短的直接证明证明了C*-代数的简单性的刻画，这个刻划最初是由Robertson和Sims提出的。
Abstract Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.