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Selfless W$^*$-probability spaces and Connes' bicentralizer problem

Cyril Houdayer, Amine Marrakchi

Abstract

We introduce the notion of selfless W$^*$-probability space and study its connection with Connes' bicentralizer problem. In particular, we show that if $M$ is a separable type ${\rm III_1}$ factor with trivial bicentralizer, then $(M, \varphi)$ is selfless for every faithful normal state $\varphi \in M_\ast$.

Selfless W$^*$-probability spaces and Connes' bicentralizer problem

Abstract

We introduce the notion of selfless W-probability space and study its connection with Connes' bicentralizer problem. In particular, we show that if is a separable type factor with trivial bicentralizer, then is selfless for every faithful normal state .

Paper Structure

This paper contains 2 sections, 2 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm-characterization']}

Key Result

Theorem 1

Let $(M, \varphi)$ be a diffuse separable $\mathrm{W}^*$-probability space. The following assertions are equivalent:

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 2
  • proof
  • Remark