Finite index subfactors and Hopf algebra crossed products

Abstract:

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