On Intermediate Subfactors[HBNI Th127]

Abstract

In this thesis we examine Pimsner-Popa bases which are not necessarily orthonormal (as in the book of Jones and Sunder) for an inclusion of subfactors of finite Jones' index and showed that this can also be done for connected inclusion of finite dimensional von Neumann algebras (in the sense that the Bratteli diagram is connected). For both these cases we obtain a characterization of `Jones' basic construction' in terms of bases and prove the phenomenon of `multistep basic construction'. In an old unpublished paper Bina Bhattacharyya and Zeph Landau has described planar algebra of an intermediate subfactor in terms of the original subfactor. In the second chapter of the thesis we have given an alternative proof of this in a more explicit and transparent manner. In the final section of this thesis we have improved the existing upper bounds for the cardinality of the finite lattice of intermediate subfactors. For this we have introduced a natural notion of `angle' between two intermediate subfactors and investigated various properties of the same. In the final section of the third chapter we investigate the intermediate subfactors for general finite-index case. We show if the norm difference between two biprojections is less than half then the corresponding intermediate subfactors are actually isomorphic.