A note on injective factors with trivial bicentralizer

TL;DR

This paper provides an alternative proof that injective factors with trivial bicentralizer are ITPFI, extending the understanding of their structure and uniqueness properties across different types.

Contribution

It offers a new proof approach for classifying injective factors with trivial bicentralizer, leveraging Haagerup's strategy and connecting to Araki-Woods' results.

Findings

Injective factors with trivial bicentralizer are ITPFI.

The proof applies uniformly across different factor types.

Uniqueness of injective factors, except type III₀, is established.

Abstract

We give an alternative proof that an injective factor on a Hilbert space with trivial bicentralizer is ITPFI. Our proof is given in parallel with each type of factors and it is based on the strategy of Haagerup. As a consequence, the uniqueness theorem of injective factors except type III$_0$ follows from Araki-Woods' result.