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 $Rinfty$. 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.
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机译：我们给出了一个由Alain Connes引起的定理的新证明，即具有可分离的前项和平凡的二集中器的III型$$$内射因子$ N $与Araki-Woods III $ 1 $因子$ R infty $同构。这与作者针对内射性III $ 1 $因子的二元集中器问题的解决方案相结合，提供了一个新的定理证明，即高达$ * $同构，在可分离的希尔伯特空间上存在一个类型为III $ 1 $的独特内射性因子。
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