How characterisations of traces and central elements fail: sharp violation constants via michelson contrast and noncentrality
Abstract
A positive functional is a trace if and only if it satisfies any one of many inequalities, the first being Gardner's |φ(X)|≤φ(|X|); positive central elements are characterised similarly. We ask how strongly these characterisations fail. For φ=Tr(ρ ·) on a matrix algebra the natural scale is the Michelson contrast δ of the density. For Gardner's inequality the sharp constant depends on δ and on the contrast δ' of |X| only through their product: it is (1−δ2δ'2)−1/2, the secant of the angle whose sine is δδ'. On a finite von Neumann algebra the contrast is the wrong parameter, since every central density gives a trace; the governing parameter is the noncentrality, defined through the central condition number κZ(ρ) of the central envelopes of ρ (the largest and smallest elements of Halpern's central spectrum). A transfer principle lifts sharp matrix constants to exact constants in terms of κZ in every finite von Neumann algebra. For 38 trace characterisations the constant is exact in 23 cases, has proved bounds in 13 and is known only numerically in two; for instance it is cosh(1/2lnκZ) for Gardner's inequality and for the subadditivity of the modulus. On matrix algebras the finite exact violations vanish at the trace to the orders δ, δ2 or δ4, and outside factors and abelian algebras no function of the contrast describes the constants. We also find the sharp constants of the Novikov-Tikhonov characterisations of central elements, sharp bounds for three unnormalised trace functionals, and sharp contrast thresholds below which subadditivity of the square root and the power-mean inequality cannot fail. A numerical search, certified as a lower bound for each supremum, supports conjectured values for the remaining inequalities.
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BibTeX
@misc{abed2026characterisations,
title = {{How characterisations of traces and central elements fail: sharp violation constants via michelson contrast and noncentrality}},
author = {Sami Abdullah Abed and Irina Nikolaeva and Andrej Novikov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34388},
eprint = {2610.34388},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34388},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - How characterisations of traces and central elements fail: sharp violation constants via michelson contrast and noncentrality
AU - Sami Abdullah Abed
AU - Irina Nikolaeva
AU - Andrej Novikov
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34388
UR - https://arxiv.org/abs/2610.34388
AB - A positive functional is a trace if and only if it satisfies any one of many inequalities, the first being Gardner's |φ(X)|≤φ(|X|); positive central elements are characterised similarly. We ask how strongly these characterisations fail. For φ=Tr(ρ ·) on a matrix algebra the natural scale is the Michelson contrast δ of the density. For Gardner's inequality the sharp constant depends on δ and on the contrast δ' of |X| only through their product: it is (1−δ^2δ'^2)^−1/2, the secant of the angle whose sine is δδ'. On a finite von Neumann algebra the contrast is the wrong parameter, since every central density gives a trace; the governing parameter is the noncentrality, defined through the central condition number κ_Z(ρ) of the central envelopes of ρ (the largest and smallest elements of Halpern's central spectrum). A transfer principle lifts sharp matrix constants to exact constants in terms of κ_Z in every finite von Neumann algebra. For 38 trace characterisations the constant is exact in 23 cases, has proved bounds in 13 and is known only numerically in two; for instance it is cosh(1/2lnκ_Z) for Gardner's inequality and for the subadditivity of the modulus. On matrix algebras the finite exact violations vanish at the trace to the orders δ, δ^2 or δ^4, and outside factors and abelian algebras no function of the contrast describes the constants. We also find the sharp constants of the Novikov-Tikhonov characterisations of central elements, sharp bounds for three unnormalised trace functionals, and sharp contrast thresholds below which subadditivity of the square root and the power-mean inequality cannot fail. A numerical search, certified as a lower bound for each supremum, supports conjectured values for the remaining inequalities.
ER -