Analyzing the Weyl Construction for Dynamical Cartan Subalgebras

Anna Duwenig; Elizabeth Gillaspy; Rachael Norton

doi:10.1093/imrn/rnab114

Analyzing the Weyl Construction for Dynamical Cartan Subalgebras

Anna Duwenig
, Elizabeth Gillaspy
, Rachael Norton

Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

When the reduced twisted C^∗-algebra C_r^∗(G, c) of a non-principal groupoid G admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of C_r^∗(G, c). In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid S of G. In this paper, we study the relationship between the original groupoids S, G and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum B of the Cartan subalgebra C_r^∗(S, c). We then show that the quotient groupoid G/S acts on B, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map G → G/S admits a continuous section, then the Weyl twist is also given by an explicit continuous 2-cocycle on G/S × B.

Original language	English
Pages (from-to)	15721-15755
Number of pages	35
Journal	International Mathematics Research Notices
Volume	2022
Issue number	20
DOIs	https://doi.org/10.1093/imrn/rnab114
State	Published - Oct 1 2022

Funding

The authors thank the referees for their very helpful comments, which have substantially streamlined and improved the paper. A.D. would further like to thank Aidan Sims and Chris Bruce for helpful discussions and the Department of Mathematics and Statistics at the University of Victoria for its hospitality. The research presented here grew out of the work [8], which was begun during the authors' participation in the 2018 BIRS workshop “Women in Operator Algebras” and continued during a 2019 stay at the Mathematical Sciences Research Institute. The latter stay was supported by the National Security Agency [H98230-19-1-0119], The Lyda Hill Foundation, The McGovern Foundation, and Microsoft Research. E.G. was partially supported by the National Science Foundation [DMS-1800749].

Funders	Funder number
DMS-1800749
National Security Agency	H98230-19-1-0119

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10.1093/imrn/rnab114

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title = "Analyzing the Weyl Construction for Dynamical Cartan Subalgebras",

abstract = "When the reduced twisted C∗-algebra Cr∗(G, c) of a non-principal groupoid G admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of Cr∗(G, c). In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid S of G. In this paper, we study the relationship between the original groupoids S, G and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum B of the Cartan subalgebra Cr∗(S, c). We then show that the quotient groupoid G/S acts on B, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map G → G/S admits a continuous section, then the Weyl twist is also given by an explicit continuous 2-cocycle on G/S × B.",

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N2 - When the reduced twisted C∗-algebra Cr∗(G, c) of a non-principal groupoid G admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of Cr∗(G, c). In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid S of G. In this paper, we study the relationship between the original groupoids S, G and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum B of the Cartan subalgebra Cr∗(S, c). We then show that the quotient groupoid G/S acts on B, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map G → G/S admits a continuous section, then the Weyl twist is also given by an explicit continuous 2-cocycle on G/S × B.

AB - When the reduced twisted C∗-algebra Cr∗(G, c) of a non-principal groupoid G admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of Cr∗(G, c). In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid S of G. In this paper, we study the relationship between the original groupoids S, G and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum B of the Cartan subalgebra Cr∗(S, c). We then show that the quotient groupoid G/S acts on B, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map G → G/S admits a continuous section, then the Weyl twist is also given by an explicit continuous 2-cocycle on G/S × B.

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JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 20

ER -

Analyzing the Weyl Construction for Dynamical Cartan Subalgebras

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