𝐾𝐾-duality for self-similar groupoid actions on graphs

IF 1.2 2区数学 Q1 MATHEMATICS

Transactions of the American Mathematical Society Pub Date : 2024-04-03 DOI:10.1090/tran/9183

Nathan Brownlowe, Alcides Buss, Daniel Gonçalves, Jeremy Hume, Aidan Sims, Michael Whittaker

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引用次数: 0

Abstract

We extend Nekrashevych’s K K KK -duality for C ∗ C^* -algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph.

More precisely, given a regular and contracting self-similar groupoid ( G , E ) (G,E) acting faithfully on a finite directed graph E E , we associate two C ∗ C^* -algebras, O ( G , E ) \mathcal {O}(G,E) and O ^ ( G , E ) \widehat {\mathcal {O}}(G,E) , to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in K K KK -theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author.

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图上自相似群元作用的 "徘徊 "对偶性

我们将内克拉舍维奇关于 C ∗ C^* -正则、递归、收缩自相似群作用的 K K KK -对偶性推广到图上的正则、收缩自相似群作用，完全去除递归条件，并从有限字母表推广到有限图。更确切地说，给定一个正则且收缩自相似的类群 ( G , E ) (G,E) 忠实地作用于有限有向图 E E，我们就会联想到两个 C ∗ C^* 算法，即 O ( G , E ) \mathcal {O}(G. E) 和 O ^ ( G , E ) \mathcal {O}(G. E) 、E) 和 O ^ ( G , E ) \widehat {mathcal {O}}(G,E) ，并证明它们强莫里塔等价于自相似极限空间的维勒孤岛所产生的斯马尔空间的稳定和不稳定的鲁埃尔 C* 算法。根据卡明克、普特南和最后一位作者证明的不可还原斯马尔空间的鲁埃尔代数的一般结果，这些代数在 K KK 理论中是斯潘尼-怀特海对偶的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Transactions of the American Mathematical Society 数学-数学

CiteScore

2.30

自引率

7.70%

发文量

171

审稿时长

3-6 weeks

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