On the Uniqueness of the Injective III₁ Factor

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.

2010 Mathematics Subject Classification: 46L36

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