2024/11/24 by Jonathan H. Brown, Brown, Jonathan H., Lisa Orloff Clark +3 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2411.15924
openalex publication_date 2024/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let D ⊆ A be a quasi-Cartan pair of algebras. Then there exists a unique discrete groupoid twist Σ→ 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⊆ C⊆ A and D ⊆ C is a quasi-Cartan pair. We also characterise which algebraic diagonal/algebraic Cartan/quasi-Cartan pairs have the property that every subalgebra C with D⊆ C⊆ A has D ⊆ 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 ⊆ A have the property that every C*-subalgebra C with D⊆ C⊆ A has D ⊆ C a Cartan pair.