Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation

1. (Dimensions)¶ The letters $d,m,n$ denote natural numbers, used as numbers of variables, and $[n]$ is the initial segment determined by $n$. $\mathbb{C}$ is the field of complex numbers, containing the real numbers $\mathbb{R}$, with imaginary unit $i$, conjugation $z\mapsto\overline{z}$, modulus $|z|$, and real and imaginary parts $\operatorname{Re}z$, $\operatorname{Im}z$.

2. (Polynomials)¶ $W_{n}$ is the set of words in the letters $1,\dots,n$, with empty word $\varnothing$. $\mathcal{P}_{n}=\mathbb{C}\langle x_{1},\dots,x_{n}\rangle$ is the algebra of noncommutative polynomials, with monomials $x_{w}$, unit $1$, variables $x_{j}$, adjoint $p\mapsto p^{*}$ and self-adjoint part $\mathcal{P}_{n,\mathrm{sa}}$. For an $n$-tuple $a$ in $\mathcal{P}_{m}$, $\sigma_{a}:\mathcal{P}_{n}\to\mathcal{P}_{m}$ is its substitution, also written $p\mapsto p(a)$.

3. (Laws)¶ $\Sigma_{n,R}$ is the set of tracial states on $\mathcal{P}_{n}$ with norm bound $R$ (for real $R>0$), and $\Sigma_{n}$ the set of noncommutative laws of $n$ variables. For a tracial state $\lambda$ on $\mathcal{P}_{n}$ and $p\in\mathcal{P}_{n}$, $\|p\|_{\lambda}$ is the nonnegative square root of the real number $\lambda(p^{*}p)\ge0$, and $M(\lambda)=\sum_{j=1}^{n}\lambda(x_{j}^{2})$, where $q^{2}=qq$. $\lambda_{k}\to\lambda$ weak-star refers to Weak-Star Convergence of Noncommutative Laws §weak-star.

4. (GNS spaces)¶ For $\lambda\in\Sigma_{n}$, $L^{2}(\lambda)$ is its GNS space, with inner product $\langle\cdot,\cdot\rangle_{\lambda}$, norm $|\cdot|_{\lambda}$, and classes $[a]_{\lambda}$ of self-adjoint polynomials $a$.

5. (Couplings and the Wasserstein distance)¶ $\iota^{1},\iota^{2}:\mathcal{P}_{d}\to\mathcal{P}_{2d}$ are the marginal substitutions; for $\mu,\nu\in\Sigma_{d}$, $\Pi(\mu,\nu)$ is the set of couplings, $\Delta_{d}$ the cost polynomial and $I(\gamma)$ the cost of a coupling; $\mu\otimes\nu$ is the tensor coupling; for a coupling $\gamma$ and $0\le t\le1$, $a^{t}$ and $\gamma_{t}$ are the tuple and the displacement interpolant of that lemma; $W_{2}$ is the noncommutative Wasserstein distance, and optimal couplings are those of The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal.

6. (Background)¶ The following results are in force and may be used without restating them: Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition, Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples, The Norm Bound of a Noncommutative Law: Multiplication by a Variable is Bounded, the Bound is Detected by Even Moments, and Laws Pull Back under Self-Adjoint Substitutions, The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It, Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound, Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants and The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation.