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

Uffe Haagerup

Vol. 137 (1989), No. 2, 265–310

DOI: 10.2140/pjm.1989.137.265

Abstract

We give a new proof for Connes’ result that an injective factor of type III_λ, 0 < λ < 1 on a separable Hilbert space is isomorphic to the Powers factor R_λ. 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,…,ζ_n) and (η₁,…,η_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 111_λ-factors: Let N be a factor of type III_λ, 0 < λ < 1, and let ϕ be a normal faithful state on N for which σ_t0^ϕ = id (t₀ = −2π∕log λ); then for every x ∈ N the norm closure of conv{uxu^∗|u ∈ U(M_ϕ)} contains a scalar operator.

Mathematical Subject Classification 2000

Primary: 46L35

Secondary: 46L10

Milestones

Received: 18 March 1988

Published: 1 April 1989