Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation. For $\gamma\in\Sigma_{2d}$ the maps $\gamma\circ\iota^{1}$ and $\gamma\circ\iota^{2}$ belong to $\Sigma_{d}$ by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §law, applied to the self-adjoint $d$-tuples $(x_{1},\dots,x_{d})$ and $(x_{d+1},\dots,x_{2d})$ of $\mathcal{P}_{2d}$, whose substitutions are the marginal substitutions $\iota^{1}$ and $\iota^{2}$. The variables are self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint. ¶For $\mu,\nu\in\Sigma_{d}$ every coupling $\gamma\in\Pi(\mu,\nu)$ lies in $\Sigma_{2d}$: choosing a real $R>0$ with $\mu,\nu\in\Sigma_{d,R}$ by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone, one has $\Pi(\mu,\nu)\subseteq\Sigma_{2d,R}$ by Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound. For $j\in[d]$ let $h_{j}=x_{j}-x_{d+j}\in\mathcal{P}_{2d}$. For a law $\lambda$ and a polynomial $p$, $\widehat{p}^{\,\lambda}$ is the class of $p$ in $\mathcal{H}_{\lambda}$; the superscript is dropped where the space is named by the subscript of the inner product.

1. (Marginal isometries)¶ For $\gamma\in\Sigma_{2d}$ and $i\in\{1,2\}$, $V^{i}_{\gamma}\in\mathcal{L}(\mathcal{H}_{\gamma\circ\iota^{i}},\mathcal{H}_{\gamma})$ is the isometry of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry for the $d$-tuple whose substitution is $\iota^{i}$; thus $V^{i}_{\gamma}\widehat{p}^{\,\gamma\circ\iota^{i}}=\widehat{\iota^{i}(p)}^{\,\gamma}$ for every $p\in\mathcal{P}_{d}$.

2. (Bounded plans)¶ Let $\mu\in\Sigma_{d}$. A bounded plan at $\mu$ is a law $\pi\in\Sigma_{2d}$ with $\pi\circ\iota^{1}=\mu$. Its momentum norm $|\pi|_{\mathrm{mom}}$ is the nonnegative square root of $\sum_{j=1}^{d}\pi(x_{d+j}x_{d+j})$, a sum of real nonnegative numbers by condition (b) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, since $x_{d+j}^{*}=x_{d+j}$.

3. (Plan pairing)¶ For $\mu\in\Sigma_{d}$, a bounded plan $\pi$ at $\mu$ and $\zeta=(\zeta_{1},\dots,\zeta_{d})\in\mathcal{H}_{\mu}^{d}$,

$$\mathcal{J}(\zeta,\pi)=\sum_{j=1}^{d}\operatorname{Re}\bigl\langle V^{1}_{\pi}\zeta_{j},\widehat{x_{d+j}}\bigr\rangle_{\mathcal{H}_{\pi}}.$$

4. (Coupling pairings)¶ For $\mu,\nu\in\Sigma_{d}$, a coupling $\gamma\in\Pi(\mu,\nu)$, $\zeta\in\mathcal{H}_{\mu}^{d}$ and $\zeta'\in\mathcal{H}_{\nu}^{d}$,

$$\mathcal{J}^{1}_{\gamma}(\zeta)=\sum_{j=1}^{d}\operatorname{Re}\bigl\langle V^{1}_{\gamma}\zeta_{j},\widehat{h_{j}}\bigr\rangle_{\mathcal{H}_{\gamma}},\qquad\mathcal{J}^{2}_{\gamma}(\zeta')=\sum_{j=1}^{d}\operatorname{Re}\bigl\langle V^{2}_{\gamma}\zeta'_{j},\widehat{h_{j}}\bigr\rangle_{\mathcal{H}_{\gamma}}.$$