Vol. 137, No. 2, 1989

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The injective factors of type IIIλ, 0 < λ < 1

Uffe Haagerup

Vol. 137 (1989), No. 2, 265–310
Abstract

We give a new proof for Connes’ result that an injective factor of type IIIλ, 0 < λ < 1 on a separable Hilbert space is isomorphic to the Powers factor Rλ. Our approach is based on lengthy, but relatively simple operations with completely positive maps together with a technical result that gives a necessary condition for that two n-tuples (ζ1,n) and (η1,n) of unit vectors in a Hilbert W-bimodule are almost unitary equivalent. As a step in the proof we obtain the following strong version of Dixmier’s approximation theorem for 111λ-factors: Let N be a factor of type IIIλ, 0 < λ < 1, and let ϕ be a normal faithful state on N for which σt0ϕ = id (t0 = 2π∕log λ); then for every x N the norm closure of conv{uxu|u U(Mϕ)} contains a scalar operator.

Mathematical Subject Classification 2000
Primary: 46L35
Secondary: 46L10
Milestones
Received: 18 March 1988
Published: 1 April 1989
Authors
Uffe Haagerup
Department of Mathematical Sciences
University of Copenhagen
Universitetsparken 5
DK-2100 Copenhagen Ø
DK-DK-2100 Copenhagen
Denmark
http://www.imada.sdu.dk/~haagerup/