The relative dimension set of an extremal subfactor of finite index

Popa’s relative dimension set \(\Lambda(M,N)\) of an inclusion of II\(_1\) factors \(N\subset M\) consists of the scalars \(\alpha\) for which some projection \(p\in M\) satisfies \(E_N(p)=\alpha 1\). 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 \(\Lambda(M,N)\) 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 \(\Lambda(M,N)\). Consequently, \(\Lambda(M,N)\) 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.

Recommended citation: Julio Cáceres. (2026). "The relative dimension set of an extremal subfactor of finite index" arXiv:2611.02336
Download Paper