A Note on Injective Factors with Trivial Bicentralizer

Abstract

We give an alternative proof that an injective factor on a Hilbert space with trivial bicentralizer is an infinite tensor product of factors of finte type I (ITPFI factor). Our proof is given in parallel with each type of factor and it is based on the strategy of Haagerup. As a consequence, the uniqueness theorem of injective factors, except for type III${}_0$, follows from Araki–Woods’ result.