Intermediate subalgebras of Cartan embeddings in rings and C*-algebras

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics Pub Date : 2025-09-17 DOI:10.1016/j.aim.2025.110534

Jonathan H. Brown , Lisa Orloff Clark , Adam H. Fuller

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引用次数: 0

Abstract

Let

D \subseteq A

be a quasi-Cartan pair of algebras. Then there exists a unique discrete groupoid twist

Σ \to G

whose twisted Steinberg algebra is isomorphic to A in a way that preserves D. In this paper, we show there is a lattice isomorphism between wide open subgroupoids of G and subalgebras C such that

D \subseteq C \subseteq A

and

D \subseteq C

is a quasi-Cartan pair. We also characterize which algebraic diagonal/algebraic Cartan/quasi-Cartan pairs have the property that every subalgebra C with

D \subseteq C \subseteq A

has

D \subseteq C

a diagonal/Cartan/quasi-Cartan pair. In the diagonal case, when the coefficient ring is a field, it is all of them. Beyond that, only pairs that are close to being diagonal have this property. We then apply our techniques to C*-algebraic inclusions and give a complete characterization of which Cartan pairs

D \subseteq A

have the property that every C*-subalgebra C with

D \subseteq C \subseteq A

has

D \subseteq C

a Cartan pair.

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环上Cartan嵌入的中间子代数与C*-代数

设D≥A为拟cartan代数对。则存在一个唯一的离散群拟扭转Σ→G，其扭曲Steinberg代数与a同构，且保持D。本文证明了G的广开子群拟与子代数C之间存在格同构，使得D、D、C是拟cartan对。我们还刻画了哪些代数对角/代数Cartan/拟Cartan对具有下述性质：每一个子代数C都有D、D、C一个对角/Cartan/拟Cartan对。在对角线的情况下，当系数环是一个场时，它是它们的全部。除此之外，只有接近对角线的对才有这个性质。然后,我们运用我们的技术C *代数夹杂物,给一个完整的描述的嘉当双D⊆属性,每一个C *子代数C与D⊆C⊆D⊆C嘉当一对。

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

Advances in Mathematics 数学-数学

CiteScore

2.80

自引率

5.90%

发文量

497

审稿时长

7.5 months

期刊介绍： Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.