Kadison-Kastler Distance and Interior Angle for Masas: Connections with Entropy and Probabilistic Index
Abstract
We study the Kadison--Kastler distance and the interior angle between masas in finite-dimensional matrix algebras. In 𝕄2(ℂ), we obtain an explicit formula for the Kadison--Kastler distance between two masas and show that every value in [0,1] is attained by dKK(Δ,uΔu*). We further prove that the distance attains its maximal value precisely when the relative unitary is a Hadamard unitary. In addition, for every v in the groupoid normaliser of Δ, the Kadison--Kastler distance between Δ and vΔv* assumes either its minimal or maximal possible value. We also establish the identity dKK(𝒜,ℬ)=sinα(𝒜,ℬ) for any two intermediate subalgebras ℂsubsetneq𝒜,ℬsubsetneq𝕄2(ℂ). Most notably, for any unitaries u,v∈𝕄2(ℂ), we establish the equivalence of maximal Kadison--Kastler distance, maximal interior angle, maximal Connes--Størmer modified entropy, and minimal Popa probabilistic index: dKK(uΔu*,vΔv*)=1 ⟺ h(uΔu*,vΔv*)=log 2 ⟺ α(uΔu*,vΔv*)=π/2 ⟺ λ(uΔu*,vΔv*)=1/2. For general 𝕄n(ℂ), we prove a unitary invariance property for the interior angle and derive an explicit formula for the angle between arbitrary masas of the form uΔ(n)u* and vΔ(n)v*. As a consequence, the interior angle is π/2 precisely when the relative unitary u*v is a Hadamard unitary. Finally, we show that every value in [0,π/2] is attained as the interior angle between a pair of masas in 𝕄n(ℂ).
Related
- Connections between the Föllmer process and the denoising diffusion probabilistic model
- The Bimetric generalization of Kastler--Kalau--Walze Type Theorems
- Rank-One Matrix Discrepancy and Algorithmic Kadison--Singer
- The solution to Kadison's problem on orthonormal bases of unitaries for type II1 factors
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BibTeX
@misc{ghosh2026kadisonkastler,
title = {{Kadison-Kastler Distance and Interior Angle for Masas: Connections with Entropy and Probabilistic Index}},
author = {Indrajit Ghosh and Sumit Kumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.38206},
eprint = {2609.38206},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38206},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Kadison-Kastler Distance and Interior Angle for Masas: Connections with Entropy and Probabilistic Index
AU - Indrajit Ghosh
AU - Sumit Kumar
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.38206
UR - https://arxiv.org/abs/2609.38206
AB - We study the Kadison--Kastler distance and the interior angle between masas in finite-dimensional matrix algebras. In 𝕄_2(ℂ), we obtain an explicit formula for the Kadison--Kastler distance between two masas and show that every value in [0,1] is attained by d_KK(Δ,uΔu^*). We further prove that the distance attains its maximal value precisely when the relative unitary is a Hadamard unitary. In addition, for every v in the groupoid normaliser of Δ, the Kadison--Kastler distance between Δ and vΔv^* assumes either its minimal or maximal possible value. We also establish the identity d_KK(𝒜,ℬ)=sinα(𝒜,ℬ) for any two intermediate subalgebras ℂsubsetneq𝒜,ℬsubsetneq𝕄_2(ℂ). Most notably, for any unitaries u,v∈𝕄_2(ℂ), we establish the equivalence of maximal Kadison--Kastler distance, maximal interior angle, maximal Connes--Størmer modified entropy, and minimal Popa probabilistic index: d_KK(uΔu^*,vΔv^*)=1 ⟺ h(uΔu^*,vΔv^*)=log 2 ⟺ α(uΔu^*,vΔv^*)=π/2 ⟺ λ(uΔu^*,vΔv^*)=1/2. For general 𝕄_n(ℂ), we prove a unitary invariance property for the interior angle and derive an explicit formula for the angle between arbitrary masas of the form uΔ^(n)u^* and vΔ^(n)v^*. As a consequence, the interior angle is π/2 precisely when the relative unitary u^*v is a Hadamard unitary. Finally, we show that every value in [0,π/2] is attained as the interior angle between a pair of masas in 𝕄_n(ℂ).
ER -