The Structure Condition at Bounded Positions for a Hamiltonian on Phase-Space Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$, with lifts $\mathcal{H}_{M}$ as in Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts; real multiples of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, differences and the $L^{2}$ norm $\lVert\cdot\rVert_{2}$ those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples, laws $\mathrm{law}(X)$ of $L^{2}$ $d$-tuples are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples, and $\kappa_{d}(\Sigma_{d,R})$ is the image of the set $\Sigma_{d,R}$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws under the canonical map $\kappa_{d}$ of Square-Integrable Noncommutative Laws: Standing Notation §laws.

¶$\mathcal{H}$ satisfies the structure condition at bounded positions if for all reals $R>0$ and $\eta>0$ there is a real $r>0$ such that, for every tracial W*-probability space $(H,M,\Omega)$, all $L^{2}$ $d$-tuples $X,Y$ of $(H,M,\Omega)$ with $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,R})$ and $\mathrm{law}(Y)\in\kappa_{d}(\Sigma_{d,R})$, and every real $\alpha>0$ with $\alpha\lVert X-Y\rVert_{2}^{2}+\lVert X-Y\rVert_{2}<r$,

$$\mathcal{H}_{M}\bigl(Y,\alpha(X-Y)\bigr)-\mathcal{H}_{M}\bigl(X,\alpha(X-Y)\bigr)<\eta.$$