The Controlled von Neumann Hamiltonian on Phase-Space Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $f:\Sigma^{2}_{d}\to\mathbb{R}$. 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 $d$-tuples $s$ in $M$, their vacuum tuples $s\Omega$, $L^{2}$ $d$-tuples, pairs $(X,Y)$ and laws $\mathrm{law}(X,Y)$ are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples; and $\mathrm{pr}^{1}_{\#}:\Sigma^{2}_{2d}\to\Sigma^{2}_{d}$ is the push-forward by the coordinate datum $\mathrm{pr}^{1}$ as in Square-Integrable Noncommutative Laws: Standing Notation §affine. For a self-adjoint $a\in M$ and $\zeta\in H$, $[\zeta,a]=JaJ\zeta-a\zeta$ is the commutator.

¶The controlled von Neumann Hamiltonian with running cost $f$ is the function $\mathcal{H}^{\mathrm{vN}}:\Sigma^{2}_{2d}\to\mathbb{R}$ defined as follows for $\pi\in\Sigma^{2}_{2d}$. If there are a tracial W*-probability space $(H,M,\Omega)$, a self-adjoint $d$-tuple $s$ in $M$ and an $L^{2}$ $d$-tuple $P$ of $(H,M,\Omega)$ with $\pi=\mathrm{law}(s\Omega,P)$, then

$$\mathcal{H}^{\mathrm{vN}}(\pi)=\frac{1}{2}\sum_{j=1}^{d}\lVert[P_{j},s_{j}]\rVert^{2}-f(\mathrm{pr}^{1}_{\#}\pi);$$

the right-hand side is the same for all such $(H,M,\Omega)$, $s$ and $P$ by The Commutator of a Square-Integrable Vector with a Bounded Self-Adjoint Operator: Linearity, the Norm Bound, Vacuum Vectors and Dependence on the Law Only §law. Otherwise $\mathcal{H}^{\mathrm{vN}}(\pi)=-f(\mathrm{pr}^{1}_{\#}\pi)$.