The Structure Condition 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, and differences and the $L^{2}$ norm $\lVert\cdot\rVert_{2}$ those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples.

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