Distances Between von Neumann Subalgebras: Spin Models, Commuting Squares, and Free Group Factors
Abstract
We investigate the relative position of von Neumann subalgebras through the Mashood--Taylor (dMT) and Kadison--Kastler (dKK) distances and the interior angle. For each n∈ℕ, we show that the hyperfinite II1-factor ℛ contains an uncountable family of pairwise distinct regular n× n spin model subfactors. In particular, this yields a continuous family (ℛHα)α∈[0,π) of 2×2 spin model subfactors which, equipped with dMT, is homeomorphic to the circle group 𝕋 and admits a natural topological group structure. We further establish dMT(ℛHα,ℛHβ) =|sin(α−β)|. We prove that two such spin model subfactors form a commuting square over their intersection if and only if they are maximally distant. More generally, commuting squares of II1-factors, under natural index conditions, force maximal distance, yielding dKK=1=dMT. We also show that diffuse subalgebras which are orthogonal in the sense of Popa are maximally distant. In contrast, no two members of the above spin model family are Popa-orthogonal. Nevertheless, maximal distance between two such subfactors implies that their interior angle over the intersection is π/2, highlighting the distinction between angle orthogonality and Popa orthogonality. Finally, in L(𝔽2)=L(⟨ a,b⟩), for u∈ L(⟨ b⟩), we obtain dMT(L(⟨ a⟩),uL(⟨ a⟩)u*) =√(1−|τ(u)|4), and construct maximally distant masas not arising from subgroups of 𝔽2.
Related
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BibTeX
@misc{ghosh2026distances,
title = {{Distances Between von Neumann Subalgebras: Spin Models, Commuting Squares, and Free Group Factors}},
author = {Indrajit Ghosh and Sumit Kumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.34856},
eprint = {2609.34856},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34856},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Distances Between von Neumann Subalgebras: Spin Models, Commuting Squares, and Free Group Factors
AU - Indrajit Ghosh
AU - Sumit Kumar
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.34856
UR - https://arxiv.org/abs/2609.34856
AB - We investigate the relative position of von Neumann subalgebras through the Mashood--Taylor (d_MT) and Kadison--Kastler (d_KK) distances and the interior angle. For each n∈ℕ, we show that the hyperfinite II_1-factor ℛ contains an uncountable family of pairwise distinct regular n× n spin model subfactors. In particular, this yields a continuous family (ℛ_H_α)_α∈[0,π) of 2×2 spin model subfactors which, equipped with d_MT, is homeomorphic to the circle group 𝕋 and admits a natural topological group structure. We further establish d_MT(ℛ_H_α,ℛ_H_β) =|sin(α−β)|. We prove that two such spin model subfactors form a commuting square over their intersection if and only if they are maximally distant. More generally, commuting squares of II_1-factors, under natural index conditions, force maximal distance, yielding d_KK=1=d_MT. We also show that diffuse subalgebras which are orthogonal in the sense of Popa are maximally distant. In contrast, no two members of the above spin model family are Popa-orthogonal. Nevertheless, maximal distance between two such subfactors implies that their interior angle over the intersection is π/2, highlighting the distinction between angle orthogonality and Popa orthogonality. Finally, in L(𝔽_2)=L(⟨ a,b⟩), for u∈ L(⟨ b⟩), we obtain d_MT(L(⟨ a⟩),uL(⟨ a⟩)u^*) =√(1−|τ(u)|^4), and construct maximally distant masas not arising from subgroups of 𝔽_2.
ER -