Plan Jets at a Common Realisation: Superdifferential and Subdifferential Momenta are Close, and the Plan Jets of a Constant

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $u:\Sigma^{2}_{d}\to\mathbb{R}$, let $\mu\in\Sigma^{2}_{d}$ and let $\delta,\delta'\ge0$ be real. The plan superjets $J^{+}_{\delta}$ and plan subjets $J^{-}_{\delta}$ are those of Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws; sums, differences, the pairing and the $L^{2}$ norm of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing; and $\widehat{M}$ and the marginal datum $\mathrm{pr}^{2}$ are those of Square-Integrable Noncommutative Laws: Standing Notation §moments and Square-Integrable Noncommutative Laws: Standing Notation §affine.

1. (Momenta at a common realisation)¶ Let $(H,M,\Omega)$ be a tracial W*-probability space and let $X,P,G$ be $L^{2}$ $d$-tuples of it with $\mathrm{law}(X,P)\in J^{+}_{\delta}u(\mu)$ and $\mathrm{law}(X,G)\in J^{-}_{\delta'}u(\mu)$. Then $\lVert P-G\rVert_{2}\le\delta+\delta'$.

2. (Plan jets of a constant)¶ Let $c\in\mathbb{R}$ and let $u$ be the constant function with value $c$. For every plan $\pi$ at $\mu$, the following are equivalent: $\pi\in J^{+}_{\delta}u(\mu)$; $\pi\in J^{-}_{\delta}u(\mu)$; $\widehat{M}(\mathrm{pr}^{2}_{\#}\pi)\le\delta^{2}$.