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 $|φ(X)|\leφ(|X|)$; positive central elements are characterised similarly. We ask how strongly these characterisations fail. For $φ=\mathrm{Tr}(ρ\,\cdot)$ 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 $\cosh(\frac12\lnκ_Z)$ 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.