Normalizers of irreducible subfactors

AbstractWe consider normalizers of an infinite index irreducible inclusion N⊆M of II1 factors. Unlike the finite index setting, an inclusion uNu∗⊆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,eN〉 is of the form u∗eNu for some unitary u∈M with uNu∗⊆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 ℓ2(G) as double cosets which are also finite unions of left cosets