Cartan subalgebras for non-principal twisted groupoid C⁎-algebras

A. Duwenig; E. Gillaspy; R. Norton; S. Reznikoff; S. Wright

doi:10.1016/j.jfa.2020.108611

Cartan subalgebras for non-principal twisted groupoid C^⁎-algebras

A. Duwenig
, E. Gillaspy
, R. Norton
, S. Reznikoff
, S. Wright

Mathematical Sciences

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

9 Scopus citations

Abstract

Renault proved in 2008 [22, Theorem 5.2] that if G is a topologically principal groupoid, then C₀(G⁽⁰⁾) is a Cartan subalgebra in C_r ^⁎(G,Σ) for any twist Σ over G. However, there are many groupoids which are not topologically principal, yet their (twisted) C^⁎-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle c on G and a subgroupoid S⊆G under which C_r ^⁎(S,c) is Cartan in C_r ^⁎(G,c). When G is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.

Original language	English
Article number	108611
Journal	Journal of Functional Analysis
Volume	279
Issue number	6
DOIs	https://doi.org/10.1016/j.jfa.2020.108611
State	Published - Oct 1 2020

Funding

This research was begun during the 2018 BIRS workshop “Women in Operator Algebras,” which was partially supported by the AWM's ADVANCE grant, and continued while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the summer of 2019. Funding for the latter stay was provided by NSA grant H98230-19-1-0119 , NSF grant 1440140 , the Lyda Hill Foundation , the McGovern Foundation , and Microsoft Research . The second author was also partially supported by NSF grant DMS-1800749 , and the fourth author was partially supported by the Simons Foundation grant 360563 .

Funders	Funder number
1440140
Simons Foundation	360563
DMS-1800749
National Security Agency	H98230-19-1-0119

Keywords

Cartan subalgebra
Groupoid 2-cocycle
Twisted groupoid C*-algebra
Weyl groupoid

Access to Document

10.1016/j.jfa.2020.108611

Cite this

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title = "Cartan subalgebras for non-principal twisted groupoid C⁎-algebras",

abstract = "Renault proved in 2008 [22, Theorem 5.2] that if G is a topologically principal groupoid, then C0(G(0)) is a Cartan subalgebra in Cr ⁎(G,Σ) for any twist Σ over G. However, there are many groupoids which are not topologically principal, yet their (twisted) C⁎-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle c on G and a subgroupoid S⊆G under which Cr ⁎(S,c) is Cartan in Cr ⁎(G,c). When G is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.",

keywords = "Cartan subalgebra, Groupoid 2-cocycle, Twisted groupoid C*-algebra, Weyl groupoid",

author = "A. Duwenig and E. Gillaspy and R. Norton and S. Reznikoff and S. Wright",

note = "Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

year = "2020",

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T1 - Cartan subalgebras for non-principal twisted groupoid C⁎-algebras

AU - Duwenig, A.

AU - Gillaspy, E.

AU - Norton, R.

AU - Reznikoff, S.

AU - Wright, S.

PY - 2020/10/1

Y1 - 2020/10/1

N2 - Renault proved in 2008 [22, Theorem 5.2] that if G is a topologically principal groupoid, then C0(G(0)) is a Cartan subalgebra in Cr ⁎(G,Σ) for any twist Σ over G. However, there are many groupoids which are not topologically principal, yet their (twisted) C⁎-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle c on G and a subgroupoid S⊆G under which Cr ⁎(S,c) is Cartan in Cr ⁎(G,c). When G is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.

AB - Renault proved in 2008 [22, Theorem 5.2] that if G is a topologically principal groupoid, then C0(G(0)) is a Cartan subalgebra in Cr ⁎(G,Σ) for any twist Σ over G. However, there are many groupoids which are not topologically principal, yet their (twisted) C⁎-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle c on G and a subgroupoid S⊆G under which Cr ⁎(S,c) is Cartan in Cr ⁎(G,c). When G is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.

KW - Cartan subalgebra

KW - Groupoid 2-cocycle

KW - Twisted groupoid C-algebra

KW - Weyl groupoid

UR - https://www.scopus.com/pages/publications/85084368984

U2 - 10.1016/j.jfa.2020.108611

DO - 10.1016/j.jfa.2020.108611

M3 - Article

AN - SCOPUS:85084368984

SN - 0022-1236

VL - 279

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 6

M1 - 108611

ER -

Cartan subalgebras for non-principal twisted groupoid C^⁎-algebras

Abstract

Funding

Keywords

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Corrigendum to “Cartan subalgebras for non-principal twisted groupoid C^⁎-algebras” (Journal of Functional Analysis (2020) 279(6), (S0022123620301543), (10.1016/j.jfa.2020.108611))

Cite this

Cartan subalgebras for non-principal twisted groupoid C⁎-algebras

Abstract

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Research output

Corrigendum to “Cartan subalgebras for non-principal twisted groupoid C⁎-algebras” (Journal of Functional Analysis (2020) 279(6), (S0022123620301543), (10.1016/j.jfa.2020.108611))

Cite this

Cartan subalgebras for non-principal twisted groupoid C^⁎-algebras

Corrigendum to “Cartan subalgebras for non-principal twisted groupoid C^⁎-algebras” (Journal of Functional Analysis (2020) 279(6), (S0022123620301543), (10.1016/j.jfa.2020.108611))