index of the subfactor

Question

If $N\subset M$ is an inclusion of type II$_1$ factors, Jones define the index $[M:N]$.

If $N$ is the type II subfactor of the type III factor $M$. How to define $[M:N]$?

Answer 1

In general, the index of an inclusion of factors depends on the conditional expectation (if it exists). There is the notion of minimal index, but this does no necessarily coincide with the Jones index for $\mathrm{II}_1$ factor inclusions.

In your case, if $E\colon M\to N$ is a normal conditional expectation, then $$ \mathrm{Ind}(E)=\inf\{\lambda\geq 0\mid \lambda E(x)\geq x\text{ for all }x\in M_+\}. $$ This version of the index is called the Pimsner-Popa index. There is another definition that relies on spatial derivatives and operator-valued weights, which coincides with the Pimsner-Popa index in this case. However, it is more complicated to spell out. A detailed account of index theory beyond the type $\mathrm{II}_1$ case is given in these lecture notes by Kosaki: https://pages.uoregon.edu/njp/lec-f.pdf.