Skew fields of non-commutative meromorphic functions affiliated to finite operator algebras
Abstract
We embed skew fields of meromorphic functions in several non-commuting variables into the topological *−algebra of closed linear operators affiliated to a finite von Neumann algebra, M, with faithful, normal tracial state, τ. Namely, let 𝒪 d (r), r ∈ (0, +∞], denote the rings of formal power series in d ∈ ℕ non-commuting variables with "radius of convergence" at least r and let Ξ∈ M 1× d be a tuple of row-norm <r with the property that there exist no non-zero trace-class operators T1, ⋯, Td ∈ ℬ (L2 (M, τ)) obeying ∑ j [Tj , Ξj] =0. Then the unital homomorphism 𝒪 d (r) → M defined by the evaluation, z j ↦ Ξj, extends to an embedding of the universal skew field of fractions of 𝒪 d (r) into the *−algebra of closed operators affiliated to M. In their L2−homology theory of von Neumann algebras, Connes and Shlyakhtenko defined a certain quantity, 0 ≤ Δ(Ξ) ≤ d, as the Murray-von Neumann co-dimension of the Hilbert module generated by all d−tuples of finite rank operators obeying the above commutator condition with Ξ∈ M 1× d. As demonstrated by Mai, Speicher and Yin, this Δ measures, in a certain precise sense, the algebraic independence of the components of Ξ. Connes-Shlyakhtenko proved that Δ is an upper bound for Voiculescu's non-microstates free entropy dimension, δ* (Ξ), and remark that it is of interest to determine whether these two quantities are equal. Using non-commutative function theory, we show that this is generally not the case, by constructing a triple of self-adjoint operators, Ξ= (Ξ1,Ξ2,Ξ3), for which this inequality is strict, δ*(Ξ) < Δ(Ξ).
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BibTeX
@misc{augat2026skew,
title = {{Skew fields of non-commutative meromorphic functions affiliated to finite operator algebras}},
author = {Méric L. Augat and Robert T. W. Martin and Maximilian Tornes and Eli Shamovich},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21004},
eprint = {2610.21004},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21004},
}
RIS
TY - UNPB
TI - Skew fields of non-commutative meromorphic functions affiliated to finite operator algebras
AU - Méric L. Augat
AU - Robert T. W. Martin
AU - Maximilian Tornes
AU - Eli Shamovich
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.21004
UR - https://arxiv.org/abs/2610.21004
AB - We embed skew fields of meromorphic functions in several non-commuting variables into the topological *−algebra of closed linear operators affiliated to a finite von Neumann algebra, M, with faithful, normal tracial state, τ. Namely, let 𝒪 _d (r), r ∈ (0, +∞], denote the rings of formal power series in d ∈ ℕ non-commuting variables with "radius of convergence" at least r and let Ξ∈ M ^1× d be a tuple of row-norm <r with the property that there exist no non-zero trace-class operators T_1, ⋯, T_d ∈ ℬ (L^2 (M, τ)) obeying ∑ _j [T_j , Ξ_j] =0. Then the unital homomorphism 𝒪 _d (r) → M defined by the evaluation, z _j ↦ Ξ_j, extends to an embedding of the universal skew field of fractions of 𝒪 _d (r) into the *−algebra of closed operators affiliated to M. In their L^2−homology theory of von Neumann algebras, Connes and Shlyakhtenko defined a certain quantity, 0 ≤ Δ(Ξ) ≤ d, as the Murray-von Neumann co-dimension of the Hilbert module generated by all d−tuples of finite rank operators obeying the above commutator condition with Ξ∈ M ^1× d. As demonstrated by Mai, Speicher and Yin, this Δ measures, in a certain precise sense, the algebraic independence of the components of Ξ. Connes-Shlyakhtenko proved that Δ is an upper bound for Voiculescu's non-microstates free entropy dimension, δ^* (Ξ), and remark that it is of interest to determine whether these two quantities are equal. Using non-commutative function theory, we show that this is generally not the case, by constructing a triple of self-adjoint operators, Ξ= (Ξ_1,Ξ_2,Ξ_3), for which this inequality is strict, δ^*(Ξ) < Δ(Ξ).
ER -