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

TL;DR

This paper introduces selfless W*-probability spaces and explores their relation to Connes' bicentralizer problem, showing that type III_1 factors with trivial bicentralizer are selfless under all faithful states.

Contribution

It establishes a new class of W*-probability spaces and links their properties to the longstanding bicentralizer problem in operator algebras.

Findings

Type III_1 factors with trivial bicentralizer are selfless for all faithful states.

Introduces the concept of selfless W*-probability spaces.

Connects selflessness with 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 $M$ is a separable type ${\rm III_1}$ factor with trivial bicentralizer, then $(M, \varphi)$ is selfless for every faithful normal state $\varphi \in M_\ast$.