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Connes' Bicentralizer Problem for Mixed $q$-deformed Araki-Woods Algebras

Panchugopal Bikram

Abstract

In this article, we show that the mixed $q$-deformed Araki-Woods von Neumann algebra $Γ_T(H_\R, U_t)^{\prime\prime}$ has trivial bicentralizer, whenever it is of type $\mathrm{III}_1$.

Connes' Bicentralizer Problem for Mixed $q$-deformed Araki-Woods Algebras

Abstract

In this article, we show that the mixed -deformed Araki-Woods von Neumann algebra has trivial bicentralizer, whenever it is of type .

Paper Structure

This paper contains 6 sections, 13 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Mixed $q$-deformed Araki-Wood von Neumann algebra
  4. Ultraproducts of von Neumann algebras
  5. Bicentralizer of $M_T$ when $\mathcal{H}_{\mathbb{R}}^{ap} \neq 0$
  6. Bicentralizer of $M_T$ when $\mathcal{H}_{\mathbb{R}}^{wm} \neq 0$

Key Result

Theorem 1.1

Let $-1<q_{ij}=q_{ji}<1$ be real numbers such that $\sup_{i,j}\left\vert q_{ij}\right\vert<1$ and let $(\mathcal{H}_{\mathbb{R}}, U_t)$ be a strongly continuous orthogonal representation such that the mixed $q$-Araki-Woods algebra $M_T$ is a type $\mathrm{III}_1$ factor, then it has trivial bicentra

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • ...and 18 more