Hamiltonians on Phase-Space Noncommutative Laws that are Uniformly Continuous on Bounded Sets

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.

¶$\mathcal{H}$ is uniformly continuous on bounded sets if for all real $R>0$ and $\eta>0$ there is a real $r>0$ such that, for every tracial W*-probability space $(H,M,\Omega)$ and all $L^{2}$ $d$-tuples $X,P,X',P'$ of $(H,M,\Omega)$ with $\lVert X\rVert_{2},\lVert P\rVert_{2},\lVert X'\rVert_{2},\lVert P'\rVert_{2}\le R$ and $\lVert X-X'\rVert_{2}+\lVert P-P'\rVert_{2}<r$,