The Carpenter property for subfactors of uniform infinite index

Abstract

We show that if $N\subset M$ is an inclusion of $\mathrm{II}_1$ factors of uniform infinite index, for instance an irreducible inclusion of infinite index, then every positive contraction $b\in N$ is of the form $E_N(p)$ for a projection $p\in M$. This gives the exact version of Popa's asymptotic result in the ultrapower, and it is a subfactor analogue of the Carpenter problem for masas. In particular, Popa's relative dimension set of such an inclusion is $[0,1]$, so the set of expectations of projections does not distinguish inclusions without corners of finite index. As applications, the Carpenter property holds for $N\subset N\rtimes G$ whenever an infinite group $G$ acts outerly on a $\mathrm{II}_1$ factor $N$, and for ICC groups $H\le G$ the inclusion $L(H)\subset L(G)$ has it exactly when $[G:H]=\infty$, confirming a recent conjecture. We also give examples showing that the Carpenter property does not imply uniform infinite index, and that infinite index does not imply the Carpenter property.