Shift-Convergent Sequences of Positions, Momenta and Shifts in a Tracial W*-Probability Space

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $r>0$ be real and let $(H,M,\Omega)$ be a tracial W*-probability space. The pairing $\langle\cdot,\cdot\rangle_{2}$ of $L^{2}$ $d$-tuples is that of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing; differences, the $L^{2}$ norm $\lVert\cdot\rVert_{2}$ and 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; convergence of real sequences is that of Limit of a Sequence of Real Numbers.

¶A shift-convergent sequence at radius $r$ in $(H,M,\Omega)$ consists of $L^{2}$ $d$-tuples $X_{n},P_{n},Q_{n}$ $(n\in\mathbb{N})$ and $X,P,Q$ of $(H,M,\Omega)$ such that: (positions) $\mathrm{law}(X_{n})\in\kappa_{d}(\Sigma_{d,r})$ for every $n$, $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,r})$, and $\lVert X_{n}-X\rVert_{2}\to0$; (momenta) $\lVert P_{n}-P\rVert_{2}\to0$; (shifts) there is a real $C$ with $\lVert Q_{n}\rVert_{2}\le C$ for every $n$, and $Q_{n}$ converges weakly to $Q$: $\langle Q_{n},Y\rangle_{2}\to\langle Q,Y\rangle_{2}$ for every $L^{2}$ $d$-tuple $Y$ of $(H,M,\Omega)$.