Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, bounded plans and the momentum norm $|\cdot|_{\mathrm{mom}}$ are those of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans, the plan pairing $\mathcal{J}$ that of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plan-pairing and the coupling pairings $\mathcal{J}^{1}_{\gamma}$ those of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing; $J^{+}$ and $J^{-}$ are the plan superjet and the plan subjet with slack $0$. Self-adjoint $d$-tuples $s$ in $M$, their vacuum tuples $s\Omega$ and their laws $\lambda_{s}$ are those of Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws, and $(s,t)$ is the self-adjoint $2d$-tuple of Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling; vacuum tuples of self-adjoint $d$-tuples are $L^{2}$ $d$-tuples by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded, and sums, differences, real multiples and the $L^{2}$ norm $\lVert\cdot\rVert_{2}$ of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing.

Let $R>0$ be real and $\mu\in\Sigma_{d,R}$; for every $k\in\mathbb{N}$ let $\nu_{k}\in\Sigma_{d,R}$ and let $\gamma_{k}\in\Pi(\mu,\nu_{k})$ be an optimal coupling; and let $(c_{k})_{k\in\mathbb{N}}$ be a sequence of nonnegative reals whose series converges. Let $(H,M,\Omega)$ be a tracial W*-probability space and let $s$ and $t^{k}$ ($k\in\mathbb{N}$) be self-adjoint $d$-tuples in $M$ with $\lVert s_{j}\rVert_{\mathrm{op}}\le R$ and $\lVert t^{k}_{j}\rVert_{\mathrm{op}}\le R$ for all $j\in[d]$ and $k\in\mathbb{N}$, $\lambda_{s}=\mu$, and $\lambda_{(s,t^{k})}=\gamma_{k}$ for every $k\in\mathbb{N}$; such data exist by Gluing Countably Many Noncommutative Couplings with a Common First Marginal in One Tracial W*-Probability Space §glue.

1. (Momentum)¶ The series $\sum_{k=1}^{\infty}c_{k}W_{2}(\mu,\nu_{k})$ converges. For every $j\in[d]$ the partial sums $\sum_{k=1}^{n}2c_{k}(s_{j}-t^{k}_{j})$ converge in operator norm, as $n\to\infty$, to a self-adjoint element $P_{j}$ of $M$; thus $P=(P_{1},\dots,P_{d})$ and $-P=(-P_{1},\dots,-P_{d})$ are self-adjoint $d$-tuples in $M$. Moreover, for every $n\in\mathbb{N}$,

$$\Bigl\lVert P\Omega-\sum_{k=1}^{n}2c_{k}\bigl(s\Omega-t^{k}\Omega\bigr)\Bigr\rVert_{2}\le2\Bigl(\sum_{k=1}^{\infty}c_{k}W_{2}(\mu,\nu_{k})-\sum_{k=1}^{n}c_{k}W_{2}(\mu,\nu_{k})\Bigr),\qquad\lVert P\Omega\rVert_{2}\le2\sum_{k=1}^{\infty}c_{k}W_{2}(\mu,\nu_{k}).$$

2. (Plans)¶ $\pi=\lambda_{(s,P)}$ and $\pi^{-}=\lambda_{(s,-P)}$ are bounded plans at $\mu$, $\kappa_{2d}(\pi)$ and $\kappa_{2d}(\pi^{-})$ are plans at $\kappa_{d}(\mu)$, and $|\pi|_{\mathrm{mom}}=|\pi^{-}|_{\mathrm{mom}}=\lVert P\Omega\rVert_{2}$.

3. (Superjet)¶ For every $\lambda\in\Sigma^{2}_{d}$ the series $\sum_{k=1}^{\infty}c_{k}\widehat{W}_{2}(\lambda,\kappa_{d}(\nu_{k}))^{2}$ converges. With $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$ its sum, $\varphi(\kappa_{d}(\mu))=\sum_{k=1}^{\infty}c_{k}W_{2}(\mu,\nu_{k})^{2}$ and $\kappa_{2d}(\pi)\in J^{+}\varphi\bigl(\kappa_{d}(\mu)\bigr)$.

4. (Subjet)¶ With $\varphi$ as in claim 3, $\kappa_{2d}(\pi^{-})\in J^{-}(-\varphi)\bigl(\kappa_{d}(\mu)\bigr)$.

5. (Pairing)¶ For every $\zeta\in\mathcal{H}_{\mu}^{d}$ the series $\sum_{k=1}^{\infty}2c_{k}\,\mathcal{J}^{1}_{\gamma_{k}}(\zeta)$ converges, and

$$\mathcal{J}(\zeta,\pi)=\sum_{k=1}^{\infty}2c_{k}\,\mathcal{J}^{1}_{\gamma_{k}}(\zeta),\qquad\mathcal{J}(\zeta,\pi^{-})=-\mathcal{J}(\zeta,\pi).$$