# On the uniqueness of the injective $\mathrm{III}_{1}$ factor

### Uffe Haagerup

## 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 *-isomorphism, there exists a unique injective factor of type III_1 on a separable Hilbert space.

## Cite this article

Uffe Haagerup, On the uniqueness of the injective $\mathrm{III}_{1}$ factor. Doc. Math. 21 (2016), pp. 1193–1226

DOI 10.4171/DM/556