The Carpenter property for subfactors of uniform infinite index
Abstract
We show that if N⊂ M is an inclusion of II1 factors of uniform infinite index, for instance an irreducible inclusion of infinite index, then every positive contraction b∈ N is of the form EN(p) for a projection p∈ 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⊂ Nrtimes G whenever an infinite group G acts outerly on a II1 factor N, and for ICC groups H≤ G the inclusion L(H)⊂ L(G) has it exactly when [G:H]=∞, 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.
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BibTeX
@misc{bisch2026carpenter,
title = {{The Carpenter property for subfactors of uniform infinite index}},
author = {Dietmar Bisch and Julio Cáceres},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02006},
eprint = {2611.02006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02006},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Carpenter property for subfactors of uniform infinite index
AU - Dietmar Bisch
AU - Julio Cáceres
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02006
UR - https://arxiv.org/abs/2611.02006
AB - We show that if N⊂ M is an inclusion of II_1 factors of uniform infinite index, for instance an irreducible inclusion of infinite index, then every positive contraction b∈ N is of the form E_N(p) for a projection p∈ 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⊂ Nrtimes G whenever an infinite group G acts outerly on a II_1 factor N, and for ICC groups H≤ G the inclusion L(H)⊂ L(G) has it exactly when [G:H]=∞, 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.
ER -