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NonHausdorff Groupoids, Proper Actions and KTheory
 DOCUMENTA MATH.
, 2004
"... Let G be a (not necessarily Hausdorff) locally compact groupoid. We introduce a notion of properness for G, which is invariant under Moritaequivalence. We show that any generalized morphism between two locally compact groupoids which satisfies some properness conditions induces a C∗correspondence ..."
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Let G be a (not necessarily Hausdorff) locally compact groupoid. We introduce a notion of properness for G, which is invariant under Moritaequivalence. We show that any generalized morphism between two locally compact groupoids which satisfies some properness conditions induces a C∗correspondence from C ∗ r(G2) to C ∗ r(G1), and thus two Morita equivalent groupoids have Moritaequivalent C∗algebras.
Deformation quantization and the BaumConnes conjecture
 Comm. Math. Phys
"... today Alternative titles of this paper would have been ‘Index theory without index’ or ‘The Baum–Connes conjecture without Baum.’ In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of C ∗algebras. We review how a wide variety of example ..."
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Cited by 5 (2 self)
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today Alternative titles of this paper would have been ‘Index theory without index’ or ‘The Baum–Connes conjecture without Baum.’ In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of C ∗algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. These include Weyl–Moyal quantization on manifolds, C ∗algebras of Lie groups and Lie groupoids, and the Etheoretic version of the Baum–Connes conjecture for smooth groupoids as described by Connes in his book Noncommutative Geometry. Concerning the latter, we use a different semidirect product construction from Connes. This enables one to formulate the Baum–Connes conjecture in terms of twisted Weyl–Moyal quantization. The underlying mechanical system is a noncommutative desingularization of a stratified Poisson space, and the Baum–Connes conjecture actually suggests a strategy for quantizing such singular spaces. 1
Quantization as a functor
 in ”Quantization, Poisson Brackets and beyond”, Contemp. Math.,315, AMS
, 2002
"... “First quantization is a mystery, but second quantization is a functor” ..."
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“First quantization is a mystery, but second quantization is a functor”
Morphisms of locally compact groupoids endowed with Haar systems
, 2005
"... We shall generalize the notion of groupoid morphism given by Zakrzewski ( [19], [20]) to the setting of locally compact σcompact Hausdorff groupoids endowed with Haar systems. To each groupoid Γ endowed with a Haar system λ we shall associate a C ∗algebra C ∗ (Γ, λ), and we construct a covariant f ..."
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We shall generalize the notion of groupoid morphism given by Zakrzewski ( [19], [20]) to the setting of locally compact σcompact Hausdorff groupoids endowed with Haar systems. To each groupoid Γ endowed with a Haar system λ we shall associate a C ∗algebra C ∗ (Γ, λ), and we construct a covariant functor (Γ, λ)→C ∗ (Γ, λ) from the category of locally compact, σcompact, Hausdorff groupoids endowed with Haar systems to the category of C ∗algebras (in the sense of [18]). If Γ is second countable and measurewise amenable, then C ∗ (Γ, λ) coincides with the full and the reduced C ∗algebras associated to Γ and λ.
A NOTION OF OPEN GENERALIZED MORPHISM THAT CARRIES AMENABILITY
"... Abstract. The purpose of this paper is to reformulate in the setting of topological groupoids the concept of morphism introduced by Zakrzewski [8]. We shall also introduce a notion of openness for these generalized morphisms and we shall prove that open generalized morphisms of locally compact Hausd ..."
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Abstract. The purpose of this paper is to reformulate in the setting of topological groupoids the concept of morphism introduced by Zakrzewski [8]. We shall also introduce a notion of openness for these generalized morphisms and we shall prove that open generalized morphisms of locally compact Hausdorff second countable groupoids carry topological amenability.