The Controlled von Neumann Hamiltonian at Bounded Positions: Evaluation, the Structure Condition and Uniform Continuity in the Momentum

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $f:\Sigma^{2}_{d}\to\mathbb{R}$, and let $\mathcal{H}^{\mathrm{vN}}$ be the controlled von Neumann Hamiltonian with running cost $f$, with lifts $\mathcal{H}^{\mathrm{vN}}_{M}$ as in Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts. Tracial W*-probability spaces $(H,M,\Omega)$ and their conjugations $J$ are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §spaces; self-adjoint tuples, vacuum tuples, $L^{2}$ tuples and their laws are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples; $\Sigma_{d,R}$ is the set of laws with norm bound $R$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws and $\kappa_{d}$ the canonical map of Square-Integrable Noncommutative Laws: Standing Notation §laws; $\lVert\cdot\rVert_{\mathrm{op}}$ is the operator norm; and $[\zeta,a]$ is the commutator.

1. (Evaluation)¶ Let $(H,M,\Omega)$ be a tracial W*-probability space, let $R>0$ be real, and let $X$ be an $L^{2}$ $d$-tuple of $(H,M,\Omega)$ with $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,R})$. Let $s$ be the self-adjoint $d$-tuple in $M$ with $s\Omega=X$ of Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §operator; thus $\lVert s_{j}\rVert_{\mathrm{op}}\le R$ for every $j\in[d]$ by Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §bound. Then for every $L^{2}$ $d$-tuple $P$ of $(H,M,\Omega)$,

$$\mathcal{H}^{\mathrm{vN}}_{M}(X,P)=\frac{1}{2}\sum_{j=1}^{d}\lVert[P_{j},s_{j}]\rVert^{2}-f(\mathrm{law}(X)).$$

2. (Structure condition)¶ If $f$ is uniformly continuous for the metric of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics, then $\mathcal{H}^{\mathrm{vN}}$ satisfies the structure condition at bounded positions.

3. (Momentum continuity)¶ $\mathcal{H}^{\mathrm{vN}}$ is uniformly continuous in the momentum at bounded positions.