Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $d,m,n\in\mathbb{N}$ and $2d=d+d$. By The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §metric, $(\Sigma_{k},W_{2})$ is a metric space for every $k\in\mathbb{N}$. Affine data $T$, their affine substitutions $\sigma_{T}$ and the coordinate data $\mathrm{pr}^{1}$, $\mathrm{pr}^{2}$, $D$ are those of Affine Data and Affine Substitutions of Noncommutative Polynomials, and $\mathrm{m}_{i}$, $\mathrm{m}_{ij}$ are the first and quadratic moments of tracial states. The real line with the absolute-value metric is a metric space by The Absolute Value Metric on the Real Line, and it is complete by Every Cauchy Sequence of Real Numbers Converges.

1. ($L^{2}$ laws)¶ For every $k\in\mathbb{N}$, the set $\Sigma^{2}_{k}$ of $L^{2}$ laws of $k$ variables is the metric completion of $(\Sigma_{k},W_{2})$. Its metric is written $\widehat{W}_{2}$, and its canonical map is written $\kappa_{k}:\Sigma_{k}\to\Sigma^{2}_{k}$.

2. (Push-forwards)¶ For an affine datum $T$ from $m$ to $n$ variables, the push-forward $T_{\#}:\Sigma^{2}_{m}\to\Sigma^{2}_{n}$ is the extension of the map $\Sigma_{m}\to\Sigma^{2}_{n}$, $\lambda\mapsto\kappa_{n}(\lambda\circ\sigma_{T})$. This map is defined by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint, it maps Cauchy sequences to Cauchy sequences by Wasserstein Estimates for Affine Push-Forwards, First and Quadratic Moments, and the Cost of a Joint Law §cauchy and The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §isometry, and its target $\Sigma^{2}_{n}$ is complete by The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §complete.

3. (Moments)¶ For $i,j\in[d]$, the first and quadratic moments $\mathrm{m}_{i},\mathrm{m}_{ij}:\Sigma^{2}_{d}\to\mathbb{R}$ are the extensions of the maps $\lambda\mapsto\mathrm{m}_{i}(\lambda)$ and $\lambda\mapsto\mathrm{m}_{ij}(\lambda)$ on $\Sigma_{d}$, which are real-valued by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §moments and map Cauchy sequences to Cauchy sequences by Wasserstein Estimates for Affine Push-Forwards, First and Quadratic Moments, and the Cost of a Joint Law §cauchy. The second moment of $\mu\in\Sigma^{2}_{d}$ is $\widehat{M}(\mu)=\sum_{i=1}^{d}\mathrm{m}_{ii}(\mu)$.

4. (Couplings)¶ For $\mu,\nu\in\Sigma^{2}_{d}$, the set of $L^{2}$ couplings of $\mu$ and $\nu$ is

5. (Cost)¶ The cost of $\gamma\in\Sigma^{2}_{2d}$ is $\mathcal{I}(\gamma)=\widehat{M}(D_{\#}\gamma)$.