Relative dimension, towers of projections and commuting squares of subfactors

Sorin Popa

Vol. 137 (1989), No. 1, 181–207

DOI: 10.2140/pjm.1989.137.181

Abstract

We study the set of projections of the type II₁ factor M which expected on the subfactor N ⊂ M are scalar multiples of the identity. The set of all these scalars, denoted Λ(M,N), is an invariant for the inclusion N ⊂ M. We compute Λ(M,N) when [M : N] < 4, when N is locally trivial and some parts of Λ(M,N) when [M : N] > 4. We prove that projections expected on the same scalar in N are conjugate by a unitary element in N. We apply all these to the commuting square problem.

Mathematical Subject Classification 2000

Primary: 46L35

Milestones

Received: 29 August 1988

Published: 1 March 1989