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## Continuous representations of groupoids (2007)

Citations: | 14 - 0 self |

### Citations

2975 | Noncommutative geometry
- Connes
- 1994
(Show Context)
Citation Context ...K-theory. Groupoid equivariant KK-theory is needed in the groupoid version of the BaumConnes conjecture [14]. Moreover, it can be used in a theoretical framework for index theorems on foliations (cf. =-=[6]-=-). A groupoid equivariant KK-cycle consists of a representation of a groupoid on a continuous field of Hilbert spaces endowed with other structure (which does not concern us here) (cf. [14]). 2000 Mat... |

162 |
Representations of ∗-Algebras, Locally Compact Groups,
- Fell, Doran
- 1988
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Citation Context ...etry, as they occur as Hilbert C∗-modules of commutative C∗-algebras (cf. Theorem 2.4). Basic sources on continuous fields of Hilbert spaces are Dixmier and Douady [8] and also work by Fell (cf. e.g. =-=[10]-=-). The main reason of our interest in representations of groupoids on continuous fields of Hilbert spaces is because of their rôle in groupoid equivariant KK-theory. Groupoid equivariant KK-theory is... |

133 |
Loop groups, Oxford Mathematical Monographs, The Clarendon Press,
- Pressley, Segal
- 1986
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Citation Context ...group of sections with the pointwise multiplication. In particular, if M is the circle S1 and G a Lie group then the group of global bisections is the loop group C(S1, G) with its usual topology (cf. =-=[16]-=-). Lemma 3.21. A continuous unitary representation (pi,H,∆) of a groupoid G⇒ M canonically induces a continuous isometric representation of Bis(G) on ∆. Proof. Define the representation p̃i of Bis(G) ... |

99 |
Sur la théorie noncommutative de l’intégration
- Connes
- 1979
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Citation Context ...piL(g)ξ(s(g)) = η(s(g)). There exist ξ′, η′ ∈ Cc(G) such that ‖η − η′‖L̂2 < ε/3, ‖ξ − ξ′‖L̂2 < ε/3 and piL(g)ξ′(s(g)) = η′(s(g)). To continue we first need the following lemma: Lemma 3.12. (A. Connes =-=[5]-=-) If f is a compactly supported continuous function on G(2), then the map g 7→ ∫ h∈Gs(g) f(g, h)λs(g)(dh) is continuous on G. Now, apply this lemma to the map f(g′, h′) := |ξ′((g′)−1h′)− η′(h′)|2. As ... |

96 |
Champs continus d'espaces hilbertiens et de C*-alge bres,
- Dixmier, Douady
- 1963
(Show Context)
Citation Context ...portant rôle in noncommutative geometry, as they occur as Hilbert C∗-modules of commutative C∗-algebras (cf. Theorem 2.4). Basic sources on continuous fields of Hilbert spaces are Dixmier and Douady =-=[8]-=- and also work by Fell (cf. e.g. [10]). The main reason of our interest in representations of groupoids on continuous fields of Hilbert spaces is because of their rôle in groupoid equivariant KK-theo... |

72 |
Représentation des produits croisés d’algèbres de groupoïdes
- Renault
- 1987
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Citation Context ...oid endowed with a Haar system {λm}m∈M , which disintegrates as λm = ∫ n∈t(Gm) λ n mµm(dn), for a Haar system {µm}m∈M on RG ⇒ M and a continuous family of measures {λnm}(n,m)∈RG on t× s : G→M ×M (cf. =-=[19]-=-). Using the family {λnm}(n,m)∈RG one can construct the continuous field of Hilbert spaces (L̂2(G),∆2(G)) := (L̂2t×s(G),∆ 2 t×s(G)), over RG, cf. Example 2.5. 838 ROGIER BOS Example 4.12. A simple exa... |

71 | Twisted K-theory of differentiable stacks
- Tu, Xu, et al.
(Show Context)
Citation Context ...m′))‖c(t(g))λm′(dg) and local boundedness of g 7→ ‖pi(g)‖. This finishes the proof. 820 ROGIER BOS A representation (H,∆, pi) is locally trivial if the continuous field (H,∆) is locally trivial. In =-=[22]-=- locally trivial representations of a groupoid G⇒M are called G-vector bundles. Representations of transitive groupoids are locally trivial. 3.2. Continuity of representations in the operator norm. In... |

62 |
Introduction to foliations and Lie groupoids
- Moerdijk, Mrčun
- 2003
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Citation Context ...y representation of a proper groupoid is isomorphic to a unitary representation, generalizing a similar result for compact groups. For groupoids we use the terminology and notational conventions from =-=[15]-=- and [18]. Suppose G⇒M a continuous groupoid, i.e. G and M are topological spaces and all structure maps are continuous. We shall assume throughout that both M and G are locally compact and Hausdorff,... |

56 |
conjecture de Novikov pour les feuilletages hyperboliques, K-Theory 16
- Tu, La
- 1999
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Citation Context ...hen any continuous representation (H,∆, pi) is isomorphic to a unitary representation. Proof. Suppose (H,∆, pi) is a non-zero continuous representation of G. Let c : M → R>0 be a cutoff function (cf. =-=[21]-=-)). This exists, since G⇒M is proper. CONTINUOUS REPRESENTATIONS OF GROUPOIDS 819 Define an inner product 〈., .〉new on H by the following description: for all m ∈M and h, h′ ∈ Hm, 〈h, h′〉new (m) := ∫ ... |

45 |
Théorie de Kasparov équivariante et groupoïdes I
- Gall
- 1999
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Citation Context ...poids on continuous fields of Hilbert spaces is because of their rôle in groupoid equivariant KK-theory. Groupoid equivariant KK-theory is needed in the groupoid version of the BaumConnes conjecture =-=[14]-=-. Moreover, it can be used in a theoretical framework for index theorems on foliations (cf. [6]). A groupoid equivariant KK-cycle consists of a representation of a groupoid on a continuous field of Hi... |

19 | Groupoid Representations for Sheaves on Orbifolds, - Pronk - 1995 |

19 |
A groupoid approach to C*-algebras, Lecture Notes in Math. 793
- Renault
- 1980
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Citation Context ...e suitable setting for a Tanaka Theorem for groupoids is that of representations on continuous (even smooth) fields of Hilbert spaces. Our study should be contrasted with the standard work of Renault =-=[18]-=- on measured representations of groupoids on measured fields of Hilbert spaces. His approach involves the construction of a quasi-invariant measure on the base space of the groupoid. Advantage of this... |

11 |
Harmonic analysis on groupoids
- Westman
(Show Context)
Citation Context ...pose of this paper is to study some of the basic theory of continuous representations in the context of groupoids. To some extent the study of representations of groupoids was initiated by Westman in =-=[24, 25]-=-. He studied representations of locally trivial groupoids on continuous vector bundles. We shall look at representations not only on continuous vector bundles, but on continuous fields of Hilbert spac... |

9 | Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids
- Bos
(Show Context)
Citation Context ...d on a continuous field of Hilbert spaces endowed with other structure (which does not concern us here) (cf. [14]). 2000 Mathematics Subject Classification. 22A30, 46L08, 46L80. 807 808 ROGIER BOS In =-=[2, 3]-=- we study geometric quantization of families with symmetry described by a groupoid action. As remarked in this paper, one obtains a groupoid equivariant KK-cycle. The geometric quantization one obtain... |

8 | Equivariant representable K-theory
- Emerson, Meyer
(Show Context)
Citation Context ... groupoid. In [22] the K-theory of C∗r (G) is shown to be generated by the locally trivial irreducible unitary representations of G, if G ⇒ M is approximately finitely generated projective (AFGP). In =-=[9]-=- it is CONTINUOUS REPRESENTATIONS OF GROUPOIDS 843 proved that G ⇒ M is AFGP if the restriction functors are dominant as above. In the case of compact groups the fact that the K-theory of the C∗-algeb... |

7 | Tannaka-Krein duality for compact groupoids II, duality.
- Amini
- 2010
(Show Context)
Citation Context ... continuous representations of groupoids on continuous fields of Hilbert spaces. As far as we know this notion as we define it does not appear anywhere in the literature. There is a preprint by Amini =-=[1]-=-, which treats continuous representations 814 ROGIER BOS on Hilbert bundles, which is rather different from the notion of continuous field of Hilbert spaces as we use it. As for representations of gro... |

5 |
Cannas da Silva, Geometric models for noncommutative algebras
- Weinstein, A
- 1999
(Show Context)
Citation Context ...s a continuous map σ : M → G such that t ◦ σ = idM and σ̃ := s ◦ σ : M → M is a 830 ROGIER BOS homeomorphism. Denote the set of global bisections of G⇒M by Bis(G). This set has a group structure, cf. =-=[23]-=-. Moreover, it is even a topological group. Lemma 3.18. Bis(G) has the structure of a topological group in the compact-open topology. Proof. The multiplication is given by (σ1 · σ2)(m) := σ1(m)σ2(σ̃1(... |

4 | Groupoids in geometric quantization
- Bos
- 2007
(Show Context)
Citation Context ...d on a continuous field of Hilbert spaces endowed with other structure (which does not concern us here) (cf. [14]). 2000 Mathematics Subject Classification. 22A30, 46L08, 46L80. 807 808 ROGIER BOS In =-=[2, 3]-=- we study geometric quantization of families with symmetry described by a groupoid action. As remarked in this paper, one obtains a groupoid equivariant KK-cycle. The geometric quantization one obtain... |

4 |
Locally trivial Cr groupoids and their representations
- Westman
- 1967
(Show Context)
Citation Context ...pose of this paper is to study some of the basic theory of continuous representations in the context of groupoids. To some extent the study of representations of groupoids was initiated by Westman in =-=[24, 25]-=-. He studied representations of locally trivial groupoids on continuous vector bundles. We shall look at representations not only on continuous vector bundles, but on continuous fields of Hilbert spac... |

3 |
les C*-algebras et leurs representations, Deuxième édition
- Dixmier
- 1969
(Show Context)
Citation Context ...-zero unitary representations on Cn with standard inner product 〈z, z′〉 = z̄z′. Let Rep0(H) be the set consisting of just the zero representation. Let S(Cn) denote the unit sphere in Cn. Lemma 3.14. (=-=[7]-=-, 18.1.9) For every integer n ≥ 0 a subbasis for the topology on Repn(H) is given by the sets U(pi, ε,K) := {pi′ ∈ Repn(H) | max g∈K |〈h′, pi(g)h〉−〈h′, pi′(g)h〉| < ε,∀h, h′ ∈ S(Cn)}, CONTINUOUS REPRES... |

3 |
Linear analysis and representation theory, Die Grundlehren der mathematischen Wissenschaften
- Gaal
(Show Context)
Citation Context ...uity and in Section 3.2 also continuity in the operator norm. All these forms of continuity can be compared, cf. Lemma 3.4, Lemma 3.5 and Lemma 3.10, generalizing similar results for groups (cf. e.g. =-=[11]-=-). In Definition 8 we introduce the notion of a morphism of representations and we show in Proposition 3.7 that any representation of a proper groupoid is isomorphic to a unitary representation, gener... |

2 |
Orbits and multiplicities for compact groups
- Heckman
- 1982
(Show Context)
Citation Context ...ights. CONTINUOUS REPRESENTATIONS OF GROUPOIDS 841 The following argument is valid if one fixes positive root systems R+K , R + H and hence fundamental Weyl chambers C+K , C + H in a way specified in =-=[12]-=-. Suppose (piλ, V ) is an irreducible representation ofK corresponding to the dominant weight λ ∈ PK ∩C+K . One can choose any integral weight Λ ∈ q−1(λ)∩PH ∩C+H ; this set is non-empty, since q is su... |

1 | On the existence of global bisections of Lie groupoids
- Chen, Liu, et al.
(Show Context)
Citation Context ... g ∈ G is in the image of a continuous global bisection; (ii) for all compact sets K ⊂M and open sets V ⊂ G, the set⋃ σ∈U(K,V ) im(σ) ⊂ G is open. Item (i) was recently addressed for Lie groupoids in =-=[4]-=-. The author believes the second condition can also be proved for Lie groupoids. Theorem 3.22. Suppose G ⇒ M is bisectional. Then there is a bijective correspondence between continuous unitary represe... |

1 |
Representations of orbifold groupoids, ArXiv:arXiv:0709.0176
- Kalisnik
(Show Context)
Citation Context ...rivial field (consider e.g. the family of groups (Z/2Z×R)\{(−1, 0)} → R). Let us also mention two recent applications of representations of groupoids on continuous fields of Hilbert spaces. First, in =-=[13]-=- Kalǐsnik shows that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. Secondly, in [20] Trentinaglia shows that the suitable setting ... |

1 |
A Tannaka Theorem for
- Trentinaglia
(Show Context)
Citation Context ...ontinuous fields of Hilbert spaces. First, in [13] Kalǐsnik shows that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. Secondly, in =-=[20]-=- Trentinaglia shows that the suitable setting for a Tanaka Theorem for groupoids is that of representations on continuous (even smooth) fields of Hilbert spaces. Our study should be contrasted with th... |