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

Each result cited is universally quantified over the data in its own statement. By The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure §lift, for every tracial W*-probability space $(H,M,\Omega)$ and all $L^{2}$ $d$-tuples $X,P$ of it,

The pairing is the inner product of $H^{d}$ and $\lVert\cdot\rVert_{2}$ its norm (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing), so Cauchy--Schwarz and the triangle inequality hold; and $\lVert Z\rVert_{2}\le\sum_{i}\lVert Z_{i}\rVert\le d\lVert Z\rVert_{2}$ for an $L^{2}$ $d$-tuple $Z$, while $\lVert\Omega\rVert=1$.

Claim 2. For $P=0$ the first two terms vanish, so $|\mathcal{H}^{\mathrm{LQ}}_{M}(X,0)|=|f(\mathrm{law}(X))|\le K$.

Claim 1. Let $R>0$ and $\eta>0$. Let $X,P,X',P'$ be $L^{2}$ $d$-tuples of one tracial W*-probability space with all four $L^{2}$ norms at most $R$, and write $\mu=\mathrm{law}(X)$, $\mu'=\mathrm{law}(X')$, $\Delta=\lVert X-X'\rVert_{2}+\lVert P-P'\rVert_{2}$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz, $\widehat{W}_{2}(\mu,\mu')\le\Delta$.

Affine images. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, $(b_{\mu}X)_{i}=c(\mu)_{i}\Omega+\sum_{j}A(\mu)_{ij}X_{j}$. With the bounds $|A(\mu)_{ij}|,|c(\mu)_{i}|\le a$ this gives $\lVert b_{\mu}X\rVert_{2}\le d\,a(1+dR)$, and, writing $b_{\mu}X-b_{\mu'}X'$ entrywise as $(c(\mu)_{i}-c(\mu')_{i})\Omega+\sum_{j}A(\mu)_{ij}(X_{j}-X'_{j})+\sum_{j}(A(\mu)_{ij}-A(\mu')_{ij})X'_{j}$ and using the Lipschitz bounds on $A$ and $c$,

Estimate. Since $\tfrac12\lVert P\rVert_{2}^{2}-\tfrac12\lVert P'\rVert_{2}^{2}=\tfrac12\langle P-P',P+P'\rangle_{2}$ and $\langle b_{\mu}X,P\rangle_{2}-\langle b_{\mu'}X',P'\rangle_{2}=\langle b_{\mu}X,P-P'\rangle_{2}+\langle b_{\mu}X-b_{\mu'}X',P'\rangle_{2}$,

By Uniformly Continuous Map Between Metric Spaces there is $r_{f}>0$ with $|f(\nu)-f(\nu')|<\eta/2$ whenever $\widehat{W}_{2}(\nu,\nu')<r_{f}$. Put $r=\min\{r_{f},\eta/(2C+1)\}$. If $\Delta<r$, then $\widehat{W}_{2}(\mu,\mu')<r_{f}$ and the right side is less than $\eta/2+\eta/2=\eta$. This is Hamiltonians on Phase-Space Noncommutative Laws that are Uniformly Continuous on Bounded Sets §uniform.