On the uniqueness of injective III$_1$ factor

Uffe Haagerup

On the uniqueness of injective III$_1$ factor

Operator Algebras 2016-06-13 v1

Authors: 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.

Keywords

injective factor type III$_1$ factor bicentralizer Araki-Woods factor *-isomorphism von Neumann algebras

Cite

@article{arxiv.1606.03156,
  title  = {On the uniqueness of injective III$_1$ factor},
  author = {Uffe Haagerup},
  journal= {arXiv preprint arXiv:1606.03156},
  year   = {2016}
}

Comments

This article is based on Uffe Haagerup's unpublished manuscript from May 1985

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