arXiv math/04084332004-08math.OAmath.DS

Mauldin-Williams graphs, Morita Equivalence and Isomorphisms

Marius Ionescu

备注：: 10 page, Latex; Corrected typos

英文摘要：

We describe a method for associating some non-self-adjoint algebras to Mauldin-Williams graphs and we study the Morita equivalence and isomorphism of these algebras. We also investigate the relationship between the Morita equivalence and isomorphism class of the C^{\ast} -correspondences associated with Mauldin-Williams graphs and the dynamical properties of the Mauldin-Williams graphs.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

相关论文

$C^{\ast}$-Algebras associated with Mauldin-Williams Graphs

A Mauldin-Williams graph $\mathcal{M}$ is a generalization of an iterated function system by a directed graph. Its invariant set $K$ plays the role of the self-similar set. We associate a $C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ with a Mauldin-Williams graph $\mathcal{M}$ and the invariant set $K$ laying emphasis on the singular points. We assume that the underlying graph $G$ has no sinks and no sources. If $\mathcal{M}$ satisfies the open set condition in $K$ and $G$ is irreducible and is not a cyclic permutation, then the associated $C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ is sim…

2005-01-19math.DSmath.OA

Operator Algebras and Mauldin Williams Graphs

We describe a method for associating a $C^{*}$-correspondence to a Mauldin-Williams graph and show that the Cuntz-Pimsner algebra of this $C^{*}$-correspondence is isomorphic to the $C^{*}$-algebra of the underlying graph. In addition, we analyze certain ideals of these $C^{*}$-algebras. We also investigate Mauldin-Williams graphs and fractal $C^{*}$-algebras in the context of the Rieffel metric. This generalizes the work of Pinzari, Watatani and Yonetani. Our main result here is a {}``no go'' theorem showing that such algebras must come from the commutative setting.

2004-10-22math.OA

Bundles of C*-correspondences over directed graphs and a theorem of Ionescu

We give a short proof of a recent theorem of Ionescu which shows that the Cuntz-Pimsner C*-algebra of a certain correspondence associated to a Mauldin-Williams graph is isomorphic to the graph algebra.

2006-01-21math.OA

The Muhly-Renault-Williams theorem for Lie groupoids and its classical counterpart

A theorem of Muhly-Renault-Williams states that if two locally compact groupoids with Haar system are Morita equivalent, then their associated convolution C*-algebras are strongly Morita equivalent. We give a new proof of this theorem for Lie groupoids. Subsequently, we prove a counterpart of this theorem in Poisson geometry: If two Morita equivalent Lie groupoids are s-connected and s-simply connected, then their associated Poisson manifolds (viz. the dual bundles to their Lie algebroids) are Morita equivalent in the sense of P. Xu.

2000-08-02math-phmath.MPmath.OA

Williams猜想对流星图成立

Williams' Conjecture holds for meteor graphs

本文证明流星图的移位等价、强移位等价、Leavitt路径代数的分次Morita等价及分次K-理论模同构彼此等价，并由此建立Leavitt路径代数与图C*-代数在Morita等价意义下的对应关系。

2023-04-12math.DSmath.FAmath.OA

Williams猜想对Gelfand-Kirillov维数为3的图成立

Williams' conjecture holds for graphs of Gelfand-Kirillov dimension three

该研究针对Gelfand-Kirillov维数为3的图，给出其强移位等价的数论准则，证明移位等价与强移位等价、Leavitt路代数分次Morita等价、分次K理论保序模同构等价，并得出相关代数与范畴等价的推论，验证Williams猜想成立。

2025-04-15math.DSmath.OAmath.RA

更多相关论文收起更多论文

小图的 Williams 与分级等价猜想

Williams' and Graded Equivalence Conjectures for small graphs

本文证明了符号动力学的 Williams 猜想与 Leavitt/$C^*$-图代数的分级 Morita 等价猜想对特定“小图”成立，并确立了小图间移位等价、强移位等价及其代数分级等变 Morita 等价的充要条件。

2024-11-12math.OAmath.RA

Crossed products of locally C*-algebras and Morita equivalence

We introduce the notion of strong Morita equivalence for group actions on locally C*-algebras and prove that the crossed products associated with two strongly Morita equivalent continuous inverse limit actions of a locally compact group G on the locally C*-algebras A and B are strongly Morita equivalent. This generalizes a result of F. Combes, Proc. London Math. Soc. 49(1984) and R. E. Curto, P.S. Muhly, D. P. Williams, Proc. Amer. Soc. 90(1984).

2007-10-02math.FAmath.OA

Equivalent groupoids have Morita equivalent Steinberg algebras

Let $G$ and $H$ be Hausdorff ample groupoids and let $R$ be a commutative unital ring. We show that if $G$ and $H$ are equivalent in the sense of Muhly-Renault-Williams, then the associated Steinberg algebras of locally constant $R$-valued functions with compact support are Morita equivalent. We deduce that collapsing a ``collapsible subgraph" of a directed graph in the sense of Crisp and Gow does not change the Morita-equivalence class of the associated Leavitt path $R$-algebra, and therefore a number of graphical constructions which yield Morita equivalent $C^*$-algebras also yield Morita eq…

2013-11-14math.OAmath.RA

Equivalence and Exact Groupoids

Given two locally compact Hausdorff groupoids $G$ and $H$ and a $(G,H)$-equivalence $Z$, one can construct the associated linking groupoid $L$. This is reminiscent of the linking algebra for Morita equivalent $C^*$-algebras. Indeed, Sims and Williams reestablished Renault's equivalence theorem by realizing $C^*(L)$ as the linking algebra for $C^*(G)$ and $C^*(H)$. Since the proof that Morita equivalence preserves exactness for $C^*$-algebras depends on the linking algebra, the linking groupoid should serve the same purpose for groupoid exactness and equivalence. We exhibit such a proof here.

2014-12-17math.OA

Flow invariants in the classification of Leavitt path algebras

We analyze in the context of Leavitt path algebras some graph operations introduced in the context of symbolic dynamics by Williams, Parry and Sullivan, and Franks. We show that these operations induce Morita equivalence of the corresponding Leavitt path algebras. As a consequence we obtain our two main results: the first gives sufficient conditions for which the Leavitt path algebras in a certain class are Morita equivalent, while the second gives sufficient conditions which yield isomorphisms. We discuss a possible approach to establishing whether or not these conditions are also in fact nec…

2011-01-19math.OAmath.RA