On the uniqueness of injective III${}_1$ factor

Abstract

We give a new proof of a theorem due to Alain Connes, that an injective factor $N$ of type III${}_1$ with separable predual and with trivial bicentralizer is isomorphic to the Araki--Woods type III${}_1$ factor $R_{\infty }$. This, combined with the author's solution to the bicentralizer problem for injective III${}_1$ factors provides a new proof of the theorem that up to $\ast$-isomorphism, there exists a unique injective factor of type III${}_1$ on a separable Hilbert space.

This article is based on Uffe Haagerup's unpublished manuscript from May 1985