Hamiltonians on Phase-Space Noncommutative Laws that are Uniformly Continuous in the Momentum at Bounded Positions

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; differences and the $L^{2}$ norm $\lVert\cdot\rVert_{2}$ of $L^{2}$ $d$-tuples are 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.

¶$\mathcal{H}$ is uniformly continuous in the momentum 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 $(H,M,\Omega)$, every $L^{2}$ $d$-tuple $X$ of $(H,M,\Omega)$ with $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,R})$, and all $L^{2}$ $d$-tuples $P,P'$ of $(H,M,\Omega)$ with $\lVert P\rVert_{2}\le R$, $\lVert P'\rVert_{2}\le R$ and $\lVert P-P'\rVert_{2}<r$,

$$\bigl|\mathcal{H}_{M}(X,P)-\mathcal{H}_{M}(X,P')\bigr|<\eta.$$