Index of faithful normal conditional expectations

Abstract:

Let $E$ be a faithful normal conditional expectation of a factor $M$ onto its subfactor $N$, and the index of $E$ be denoted by ${\operatorname {IND}_E}$. We investigate the question: For two such faithful normal conditional expectations ${E_1},{E_2}$ of $M$ onto $N$, when does ${\operatorname {IND}}_{{E_1}} = {\operatorname {IND}}_{{E_2}}$ hold? In this paper we answer this question completely for type $I$ factor $M$. We also derive a tensor product formula for index, i.e., ${\operatorname {IND}}_{{E_1} \otimes {E_2}} = ({\operatorname {IND}}_{{E_1}})({\operatorname {IND}}_{{E_2}})$. For any $\alpha > 9$ we construct uncountable nonisomorphic faithful normal conditional expectations $E$ of a factor $M$ onto its subfactor $N$ with ${\operatorname {IND}_E} = \alpha$ such that both of $M$ and $N$, are of type $I$ or $II$ or $II{I_\lambda },0 \leq \lambda \leq 1$. For each $\beta \in \{ 4{\cos ^2}\pi /n,|n \geq 3\} \cup [4,\infty )$ we exhibit a type $II{I_\lambda }$ factor $M$ and its subfactor $N$ and a faithful normal conditional expectation $E$ such that ${\operatorname {IND}}_E = \beta$.