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Operator algebras of free wreath products

Pierre Fima, Arthur Troupel

Abstract

We give a description of operator algebras of free wreath products in terms of fundamental algebras of graphs of operator algebras as well as an explicit formula for the Haar state. This allows us to deduce 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, fullness, primeness and absence of Cartan subalgebra and we give a formula for Connes' $T$-invariant and $τ$-invariant. We also study maximal amenable von Neumann subalgebras. Finally, we give some explicit computations of K-theory groups for C*-algebras of free wreath products. As an application we show that the reduced C*-algebras of quantum reflection groups are pairwise non-isomorphic.

Operator algebras of free wreath products

Abstract

We give a description of operator algebras of free wreath products in terms of fundamental algebras of graphs of operator algebras as well as an explicit formula for the Haar state. This allows us to deduce 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, fullness, primeness and absence of Cartan subalgebra and we give a formula for Connes' -invariant and -invariant. We also study maximal amenable von Neumann subalgebras. Finally, we give some explicit computations of K-theory groups for C*-algebras of free wreath products. As an application we show that the reduced C*-algebras of quantum reflection groups are pairwise non-isomorphic.
Paper Structure (19 sections, 38 theorems, 74 equations)

This paper contains 19 sections, 38 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Full von Neumann algebras
  5. Compact quantum groups
  6. Free product quantum group
  7. Semi-direct product quantum group
  8. Quantum permutation group
  9. Free wreath products
  10. Graphs of operator algebras
  11. Free wreath products as fundamental algebras
  12. The full version
  13. The reduced version
  14. The von Neumann version
  15. An inductive construction for free wreath products algebras
  16. ...and 4 more sections

Key Result

Theorem A

Given $N\in\mathbb{N}^*$ and $G$ a compact quantum group. The following holds.

Theorems & Definitions (90)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 80 more