Normalizers of irreducible subfactors

Abstract

We consider normalizers of an infinite index irreducible inclusion N ⊆ M of II 1 factors. Unlike the finite index setting, an inclusion u N u ∗ ⊆ N can be strict, forcing us to also investigate the semigroup of one-sided normalizers. We relate these one-sided normalizers of N in M to projections in the basic construction and show that every trace one projection in the relative commutant N ′ ∩ 〈 M , e N 〉 is of the form u ∗ e N u for some unitary u ∈ M with u N u ∗ ⊆ N generalizing the finite index situation considered by Pimsner and Popa. We use this to show that each normalizer of a tensor product of irreducible subfactors is a tensor product of normalizers modulo a unitary. We also examine normalizers of infinite index irreducible subfactors arising from subgroup–group inclusions H ⊆ G . Here the one-sided normalizers arise from appropriate group elements modulo a unitary from L ( H ) . We are also able to identify the finite trace L ( H ) -bimodules in ℓ 2 ( G ) as double cosets which are also finite unions of left cosets.