Finite index subfactors and Hopf algebra crossed products

Abstract

We show that if N ⊆ M ⊆ L ⊆ K {\mathbf {N}} \subseteq {\mathbf {M}} \subseteq {\mathbf {L}} \subseteq {\mathbf {K}} is a Jones’s tower of type I I 1 {\text {I}}{{\text {I}}_1} factors satisfying [ M : N ] > ∞ , N ′ ∩ M = C I , N ′ ∩ K [{\mathbf {M}}:{\mathbf {N}}] > \infty ,\;{\mathbf {N}}’ \cap {\mathbf {M}} = \mathbb {C}I,\;{\mathbf {N}}’ \cap {\mathbf {K}} a factor, then M ′ ∩ K {\mathbf {M}}’ \cap {\mathbf {K}} bears a natural Hopf ∗ {\ast } -algebra structure and there is an action of M ′ ∩ K {\mathbf {M}}’ \cap {\mathbf {K}} on L {\mathbf {L}} such that the resulting crossed product is isomorphic to K {\mathbf {K}} .