The relative dimension set of an extremal subfactor of finite index
Abstract
Popa's relative dimension set Λ(M,N) of an inclusion of II1 factors N⊂ M consists of the scalars α for which some projection p∈ M satisfies EN(p)=α1. Popa computed it for index at most $4$, and for index d>4 outside the middle interval (td,1−td), where td(1−td)=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 [td,1−td] 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.
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BibTeX
@misc{caceres2026relative,
title = {{The relative dimension set of an extremal subfactor of finite index}},
author = {Julio Cáceres},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02336},
eprint = {2611.02336},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02336},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The relative dimension set of an extremal subfactor of finite index
AU - Julio Cáceres
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02336
UR - https://arxiv.org/abs/2611.02336
AB - Popa's relative dimension set Λ(M,N) of an inclusion of II_1 factors N⊂ M consists of the scalars α for which some projection p∈ M satisfies E_N(p)=α1. 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.
ER -