Intermediate Subalgebras of Cartan embeddings in rings and C*-algebras
Jonathan H. Brown, Lisa Orloff Clark, Adam H. Fuller

TL;DR
This paper explores the structure of intermediate subalgebras in Cartan embeddings within rings and C*-algebras, establishing a correspondence with subgroupoids and characterizing when subalgebras preserve Cartan properties.
Contribution
It introduces a lattice isomorphism between subgroupoids and subalgebras in quasi-Cartan pairs, and characterizes when intermediate subalgebras retain Cartan or diagonal structures.
Findings
Lattice isomorphism between subgroupoids and subalgebras in quasi-Cartan pairs
Characterization of subalgebras preserving Cartan properties in rings and C*-algebras
Complete description of Cartan pairs with all intermediate subalgebras also being Cartan
Abstract
Let be a quasi-Cartan pair of algebras. Then there exists a unique discrete groupoid twist whose twisted Steinberg algebra is isomorphic to in a way that preserves . In this paper, we show there is a lattice isomorphism between wide open subgroupoids of and subalgebras such that and is a quasi-Cartan pair. We also characterise which algebraic diagonal/algebraic Cartan/quasi-Cartan pairs have the property that every subalgebra with has 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 …
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Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Operator Algebra Research · Algebraic structures and combinatorial models
