ORCID Profile
0000-0001-6042-2561
Current Organisation
KU Leuven
Does something not look right? The information on this page has been harvested from data sources that may not be up to date. We continue to work with information providers to improve coverage and quality. To report an issue, use the Feedback Form.
Publisher: Oxford University Press (OUP)
Date: 05-07-2021
DOI: 10.1093/IMRN/RNAB114
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 ${\\mathcal{G}}/{\\mathcal{S}} \\ltimes{\\mathfrak{B}}$.
Publisher: Elsevier BV
Date: 2022
Publisher: Elsevier BV
Date: 10-2020
Publisher: American Mathematical Society (AMS)
Date: 02-12-2020
DOI: 10.1090/BTRAN/54
Abstract: An early result of Noncommutative Geometry was Connes’ observation in the 1980’s that the Dirac-Dolbeault cycle for the 2 2 -torus T 2 \\mathbb {T}^2 , which induces a Poincaré self-duality for T 2 \\mathbb {T}^2 , can be ‘quantized’ to give a spectral triple and a K-homology class in K K 0 ( A θ ⊗ A θ , C ) \\mathrm {KK}_0(A_\\theta \\otimes A_\\theta , \\mathbb {C}) providing the co-unit for a Poincaré self-duality for the irrational rotation algebra A θ A_\\theta for any θ ∈ R ∖ Q \\theta \\in \\mathbb {R}\\setminus \\mathbb {Q} . Connes’ proof, however, relied on a K-theory computation and does not supply a representative cycle for the unit of this duality. Since such representatives are vital in applications of duality, we supply such a cycle in unbounded form in this article. Our approach is to construct, for any non-zero integer b b , a finitely generated projective module L b \\mathcal {L}_{b} over A θ ⊗ A θ A_\\theta \\otimes A_\\theta by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope θ \\theta and θ + b \\theta + b , using the fact that these flows are transverse to each other. We then compute Connes’ dual of [ L b ] [\\mathcal {L}_{b}] and prove that we obtain an invertible τ b ∈ K K 0 ( A θ , A θ ) \\tau _{b}\\in \\mathrm {KK}_0(A_\\theta , A_\\theta ) , represented by an equivariant bundle of Dirac-Schrödinger operators. An application of equivariant Bott Periodicity gives a form of higher index theorem describing functoriality of such ‘ b b -twists’ and this allows us to describe the unit of Connes’ duality in terms of a combination of two constructions in KK-theory. This results in an explicit spectral representative of the unit – a kind of ‘quantized Thom class’ for the diagonal embedding of the noncommutative torus.
Publisher: Mathematisches Forschungsinstitut Oberwolfach
Date: 2022
DOI: 10.14760/OWR-2022-36
Publisher: Rocky Mountain Mathematics Consortium
Date: 02-2020
Publisher: Elsevier BV
Date: 12-2022
No related grants have been discovered for Anna Duwenig.