The Linear-Quadratic Hamiltonian is Uniformly Continuous on Bounded Sets and Bounded at Zero Momentum

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\rho$, $f$, the affine data $b_{\mu}$ ($\mu\in\Sigma^{2}_{d}$) and $\mathcal{H}^{\mathrm{LQ}}$ be as in The Linear-Quadratic Hamilton-Jacobi Equation with Law-Dependent Affine Drift on Square-Integrable Noncommutative Laws, and assume the hypotheses of The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure, with its reals $a,L\ge0$. Lifts are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, and $0=(0,\dots,0)$ is the zero $L^{2}$ $d$-tuple.

1. (Uniform continuity)¶ $\mathcal{H}^{\mathrm{LQ}}$ is uniformly continuous on bounded sets.

2. (Zero momentum)¶ Let $K\in\mathbb{R}$ satisfy $|f(\nu)|\le K$ for every $\nu\in\Sigma^{2}_{d}$. Then $|\mathcal{H}^{\mathrm{LQ}}_{M}(X,0)|\le K$ for every tracial W*-probability space $(H,M,\Omega)$ and every $L^{2}$ $d$-tuple $X$ of it.