Square-Integrable Noncommutative Laws: Standing Notation

1. ($L^{2}$ laws)¶ The conventions of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation are in force. For $k\in\mathbb{N}$, $\Sigma^{2}_{k}$ is the set of $L^{2}$ laws of $k$ variables, with metric $\widehat{W}_{2}$ and canonical map $\kappa_{k}:\Sigma_{k}\to\Sigma^{2}_{k}$; $2d=d+d$.

2. (Affine data)¶ For $m,n\in\mathbb{N}$, affine data $T=(A,c)$ from $m$ to $n$ variables, their affine substitutions $\sigma_{T}$ and norms $\lVert T\rVert$, composites $S\circ T$, identity data $\mathrm{id}_{m}$ and the coordinate data $\mathrm{pr}^{1},\mathrm{pr}^{2},D,\mathrm{diag}$ are those of Affine Data and Affine Substitutions of Noncommutative Polynomials, and $T_{\#}:\Sigma^{2}_{m}\to\Sigma^{2}_{n}$ is the push-forward.

3. (Moments, couplings and cost)¶ $\mathrm{m}_{i}$ and $\mathrm{m}_{ij}$ are the first and quadratic moments, of tracial states as in Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §moments and of $L^{2}$ laws as in Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §moments; $\widehat{M}$ is the second moment; $\Pi^{2}(\mu,\nu)$ is the set of $L^{2}$ couplings of $\mu,\nu\in\Sigma^{2}_{d}$ and $\mathcal{I}$ the cost.

4. (Background)¶ The following results are in force and may be used without restating them: The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It, Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data, Wasserstein Estimates for Affine Push-Forwards, First and Quadratic Moments, and the Cost of a Joint Law, Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling and The Distance between Square-Integrable Noncommutative Laws is the Infimum of the Cost over Their Couplings.