Generalized free wreath products and their operator algebras

Abstract: We develop a new approach on free wreath products, generalizing the constructions of Bichon and of Fima-Pittau. We show stability properties for certain approximation properties such as exactness, Haagerup property, hyperlinearity and K-amenability. We study qualitative properties of the associated von Neumann algebra: factoriality, primeness and absence of Cartan subalgebra and we give a formula for Connes' T-invariant. Finally, we give some explicit computations of K-theory groups for C*-algebras of generalized free wreath products.