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

Abstract

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