Mimicking a recent article of Stefaan Vaes, in which was proved that every locally compact quantum group can act outerly, we prove that we get the same result for measured quantum groupoids, with an appropriate definition of outer actions of measured quantum groupoids. This result is used to show that every measured quantum groupoid can be found from some depth 2 inclusion of von Neumann algebras.
We classify outer actions (or $\mathscr{G}$-kernels) of discrete amenable groupoids on injective factors. Our method based on unified approach for classification of discrete amenable groups actions, and cohomology reduction theorem of discrete amenable equivalence relations. We do not use Katayama-Takesaki type resolution group approach.
The theory of measured quantum groupoids, as defined by Lesieur and myself, was made to generalize the theory of quantum groups made by Kustarmans and Vaes, but was only defined in a von Neumann algebra setting; Th. Timmermann constructed locally compact quantum groupoids, which is a C*-version of quantum groupoids. Here, we associate to such a locally compact quantum groupoid a measured quantum groupoid in which it is weakly dense; we then associate to a measured quantum groupoid a locally compact quantum groupoid which is weakly dense in the measured quantum groupoid, but such a locally comp…
弗兰克·勒庞在他的论文中介绍了一个量子广群的概念（现在发表在{ \it Mémoires de la SMF}（2007）的扩展和修订版中），在冯·诺依曼代数的背景下，勒庞公理的简化在本文的附录中给出。我们在这里发展的概念的作用，交叉产品，并获得一个双对偶定理，已完成的Stefaan Vaes为局部紧量子群。此外，我们还证明了初始代数在其交叉积中的包含深度为2，这是Jean-Michel Vallin和作者证明的一个结果的逆.更准确地说，对于一个可测量子广群的任何作用，我们都可以将另一个可测量子广群联系起来。特别地，我们从局部紧量子群的作用出发，得到了一个与该作用正则相关的可测量子群胚;当该作用是外作用时，这个可测量子群胚就是初始的局部紧量子群
In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning actions of locally compact quantum groups on C*-algebras [S. Baaj, G. Skandalis and S. Vaes, 2003]. Let $\cal G$ be a measured quantum groupoid on a finite basis. We prove that if $\cal G$ is regular, then any weakly continuous action of $\cal G$ on a C*-algebra is necessarily strongly continuous. Following [S. Baaj and G. Skandalis, 1989], we introduce and investigate a notion of $\cal G$-equivariant Hilbert C$^*$-modules. By applying the previous results and a version …
In this article, when G is a locally compact quantum group, we associate to a braided-commutative G-Yetter-Drinfel'd algebra $(N,a,\hat{a})$ equipped with a normal faithful semi-finite weight verifying some appropriate condition, a structure of a measured quantum groupoid. The dual structure is then given by $(N,\hat{a},a)$. Examples are given, especially the situation of a quotient type co-ideal of a compact quantum group. This construction generalizes the standard construction of a transformation groupoid. Most of the results were announced by the second author in 2011, at a conference in Wa…
In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning equivariant Kasparov theory for actions of locally compact quantum groups [S. Baaj and G. Skandalis, 1989, 1993]. To every pair $(A,B)$ of C*-algebras continuously acted upon by a regular measured quantum groupoid on a finite basis $\cal G$, we associate a $\cal G$-equivariant Kasparov theory group ${\sf KK}_{\cal G}(A,B)$. The Kasparov product generalizes to this setting. By applying recent results concerning actions of regular measured quantum groupoids on a finite bas…
In this article, we generalize to the case of regular locally compact quantum groups, two important results concerning actions of compact quantum groups. Let $G_1$ and $G_2$ be two monoidally equivalent regular locally compact quantum groups in the sense of De Commer. We introduce an induction procedure and we build an equivalence of the categories ${A}^{G_1}$ and ${A}^{G_2}$ consisting of continuous actions of $G_1$ and $G_2$ on $C^*$-algebras. As an application of this result, we derive a canonical equivalence of the categories ${KK}^{G_1}$ and ${KK}^{G_2}$. We introduce and investigate a no…
