Actions and coactions of finite quantum groupoids on von Neumann algebras, extensions of the matched pair procedure

Abstract

In this work, 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 an universal von Neumann algebra on which any finite groupoid acts outerly. We give two actually different extensions of the matched pairs procedure. In previous works, N. Andruskiewitsch and S. Natale define, 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 an other type of finite C ∗ -quantum groupoids.