Generalized free wreath products and their operator algebras
Pierre Fima, Arthur Troupel
TL;DR
The paper introduces a comprehensive, universal construction of generalized free wreath products $ ext{G}=G times_{*,eta,F}H$ amalgamated over a dual quantum subgroup $F$, linking and unifying prior wreath product frameworks. It develops a block-extended C*-algebraic toolkit to analyze the reduced and von Neumann algebras, establishing exactness, Haagerup, hyperlinear, and K-amenability stability criteria in terms of the factors $G$ and $H$, and their duals. It then derives detailed qualitative properties of ${ m L}^ f( ext{G})$, including factoriality, type classification, primeness, and absence of Cartan subalgebras under 2-ergodicity with infinite-dimensional $G,H$ and finite $F$, together with Connes’ T-invariant computations. Finally, the work provides KK-theoretic insights and explicit $K$-theory calculations for C*-algebras of generalized free wreath products, including concrete formulas in key cases such as $ ext{Z}_s times{ m Aut}^+(M_N(C), u)$ and free products with Aut$(M_N(C), u)$. These results extend known Bichon/Fima–Pittau cases to a broad amalgamated framework with concrete operator-algebraic consequences and computable invariants for examples arising from finite-dimensional base 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.