Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids

Abstract

Given a non-necessarily Hausdorff, topologically free, twisted étale groupoid (G, L ), we consider its essential groupoid C*-algebra, denoted C * ess (G, L ), obtained by completing Cc(G, L ) with the smallest among all C*seminorms coinciding with the uniform norm on Cc(G (0) ). The inclusion of C*-algebrasis then proven to satisfy a list of properties characterizing it as what we call a weak Cartan inclusion. We then prove that every weak Cartan inclusion A, B , with B separable, is modeled by a topologically free, twisted étale groupoid, as above. In our second main result we give a necessary and sufficient condition for an inclusion of C*-algebras A, B to be modeled by a twisted étale groupoid based on the notion of canonical states. A simplicity criterion for C * ess (G, L ) is proven and many examples are provided.

Localised Hilbert modules and weak noncommutative Cartan pairs

“…Under this definition, C ess pG, Aq is the quotient of C ˚pG, Aq determined by the canonical local expectation EL : C ˚pG, Aq -A ¸S Ñ M loc pAq, and we can moreover consider C ess pG, Aq as a quotient of C r pG, Aq using Proposition 6.2. Kwaśniewski and Meyer call elements of the kernel of the map C r pG, Aq Ñ C ess pG, Aq singular, following Exel and Pitts [9] and denote the kernel of this map by J sing . There is also by [14,Proposition 7.9] an injective norm-decreasing homomorphism j : C r pG, Aq Ñ BpG, Aq, whereby we can consider elements of C r pG, Aq as (Borel) sections G Ñ A.…”

Dauns-Hofmann-Kumjian-Renault Duality for Fell Bundles and Structured C*-Algebras

“…This could allow the duality to be extended to Fell bundles over even non-Hausdorff groupoids. Again our previous work in [Bic20b], as well as recent work of Exel and Pitts in [EP19], shows that this is not beyond the realm of possibility, although the technical hurdles to overcome may be significant.…”

On Kumjian's C*-diagonals and the opaque ideal

“…of an abelian C*-algebra A in another C*-algebra B, satisfying suitable hypotheses, is necessarily modeled by a twisted, principal, étale groupoid. The first main goal of the present paper is to prove a souped up version of Kumjian's result, based on my recent work [7] with D. Pitts. The plan is to strip the hypotheses of [11: Theorem 3.1] to a bare minimum, while retaining its conclusion, except that the modeling will be done with a possibly exotic groupoid C*-algebra, rather than the reduced version adopted in [11].…”