1. Analyzing the Weyl Construction for Dynamical Cartan Subalgebras
- Author
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Elizabeth Gillaspy, Anna Duwenig, and Rachael Norton
- Subjects
General Mathematics ,01 natural sciences ,Section (fiber bundle) ,Combinatorics ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,46L05, 22D25, 22A22 ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Twist ,Operator Algebras (math.OA) ,Mathematics::Representation Theory ,Quotient ,Mathematics ,Science & Technology ,Mathematics::Operator Algebras ,010102 general mathematics ,Spectrum (functional analysis) ,Mathematics - Operator Algebras ,Cartan subalgebra ,C-ASTERISK-ALGEBRAS ,Physical Sciences ,010307 mathematical physics ,EQUIVALENCE - Abstract
When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages
- Published
- 2022