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

Mathematics

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

8 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

Keywords

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

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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",

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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

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DO - 10.1016/j.jfa.2020.108611

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JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

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

Abstract

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

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))