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Actions and coactions of finite quantum groupoids on von Neumann algebras, extensions of the matched pair procedure. (English) Zbl 1128.46026
Summary: Actions and coactions of finite ${C}^{*}$-quantum groupoids are studied in an operator algebras context. In particular, we prove a double crossed product theorem, and the existence of a universal von Neumann algebra on which any finite groupoid acts outerly. We give two actually different extensions of the matched pairs procedure. In [Publ. Mat. Urug. 10, 11–51 (2005; Zbl 1092.16021)], N. Andruskiewitsch and S. Natale defined, for any matched pair of groupoids, two ${C}^{*}$-quantum groupoids in duality; we give here an interpretation of them in terms of crossed products of groupoids using a single multiplicative partial isometry which gives a complete description of these structures. The second extension deals only with groups to define another type of finite ${C}^{*}$-quantum groupoids.
MSC:
 46L65 Quantizations, deformations (${C}^{*}$-algebras) 22A22 Topological groupoids (including differentiable and Lie groupoids)