Approximately inner automorphisms on inclusions of type III_λ-factors

Carl Winsløw

Vol. 166 (1994), No. 2, 385–400

DOI: 10.2140/pjm.1994.166.385

Abstract

For arbitrary inclusions of factors with finite index, we define a “fundamental homomorphism” which is a generalization of both the Connes-Takesaki fundamental homomorphism for properly infinite (single) factors and Loi’s construction for inclusions of type II₁-factors.

It is shown that for nice inclusions of type III_λ-factors (0 < λ < 1), the kernel of the fundamental homomorphism coincides with the set of approximately inner automorphisms on the inclusion. To prove this, we first give a characterization of approximate innerness on type III_λ-inclusions in terms of Loi’s and Connes-Takesaki’s invariants.

Mathematical Subject Classification 2000

Primary: 46L40

Secondary: 46L35

Milestones

Received: 30 July 1992

Accepted: 14 December 1992

Published: 1 December 1994