The relative dimension set of an extremal subfactor of finite index

Abstract

Popa's relative dimension set $Λ(M,N)$ of an inclusion of $\mathrm{II}_1$ factors $N\subset M$ consists of the scalars $α$ for which some projection $p\in M$ satisfies $E_N(p)=\alpha1$. Popa computed it for index at most $4$, and for index $d>4$ outside the middle interval $(t_d,1-t_d)$, where $t_d(1-t_d)=d^{-1}$; whether the middle interval meets $Λ(M,N)$ for every irreducible inclusion of index $d>4$ has remained open. We show that for every extremal inclusion of finite index $d>4$, the whole interval $[t_d,1-t_d]$ is contained in $Λ(M,N)$. Consequently, $Λ(M,N)$ depends only on the index for all extremal inclusions of finite index, in particular for all irreducible ones. Extremality cannot be dropped: for $d>4$, Popa's locally trivial inclusions are exactly those whose relative dimension set misses the middle interval. Other non-extremal inclusions can contain the whole middle interval in their relative dimension set, or miss part of it.