Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants

Let $\mathbb{N}$ be the set of natural numbers, $\mathbb{C}$ the field of complex numbers and $\mathbb{R}\subseteq\mathbb{C}$ the real numbers. Let $d\in\mathbb{N}$ and for $n\in\mathbb{N}$ let $\mathcal{P}_{n}=\mathbb{C}\langle x_{1},\dots,x_{n}\rangle$ be the noncommutative polynomials, with unit $1$, monomials $x_{w}$ indexed by the words $w\in W_{n}$, and substitutions $\sigma_{a}$. Let $\Sigma_{n,R}$ be the set of tracial states on $\mathcal{P}_{n}$ with norm bound $R$ and $\Sigma_{n}$ the set of noncommutative laws of $n$ variables, weak-star convergence as in Weak-Star Convergence of Noncommutative Laws §weak-star, and let $\iota^{1},\iota^{2}$, $\Pi(\mu,\nu)$ and $I$ be the marginal substitutions, the sets of couplings and the cost. For $q$ in $\mathcal{P}_{n}$ write $q^{2}=qq$, and for real $s$ write $s^{2}=ss$. For a tracial state $\lambda$ on $\mathcal{P}_{d}$ write $M(\lambda)=\sum_{j=1}^{d}\lambda(x_{j}^{2})$. Let $\mu,\nu\in\Sigma_{d}$ and let $R>0$ be real.

1. (Norm bound)¶ If $\mu,\nu\in\Sigma_{d,R}$, then $\Pi(\mu,\nu)\subseteq\Sigma_{2d,R}$.

2. (Cost)¶ $M(\mu)$ and $M(\nu)$ are real and nonnegative, and for every $\gamma\in\Pi(\mu,\nu)$ the numbers $\gamma(x_{j}x_{d+j})$ are real and

3. (Tensor coupling)¶ Let $\sigma^{1}$ and $\sigma^{2}$ be the substitutions, from $\mathcal{P}_{2d}$ to $\mathcal{P}_{d}$, of the $2d$-tuples $(x_{1},\dots,x_{d},1,\dots,1)$ and $(1,\dots,1,x_{1},\dots,x_{d})$ in $\mathcal{P}_{d}$. There is exactly one linear map $\mu\otimes\nu:\mathcal{P}_{2d}\to\mathbb{C}$ with $(\mu\otimes\nu)(x_{w})=\mu(\sigma^{1}(x_{w}))\,\nu(\sigma^{2}(x_{w}))$ for every $w\in W_{2d}$ (by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension), and $\mu\otimes\nu\in\Pi(\mu,\nu)$. In particular $\Pi(\mu,\nu)$ is nonempty.

4. (Diagonal coupling)¶ Let $\delta$ be the substitution, from $\mathcal{P}_{2d}$ to $\mathcal{P}_{d}$, of the $2d$-tuple $(x_{1},\dots,x_{d},x_{1},\dots,x_{d})$. Then $\mu\circ\delta\in\Pi(\mu,\mu)$ and $I(\mu\circ\delta)=0$.

5. (Swapped coupling)¶ Let $s$ be the substitution, from $\mathcal{P}_{2d}$ to $\mathcal{P}_{2d}$, of the $2d$-tuple $(x_{d+1},\dots,x_{2d},x_{1},\dots,x_{d})$. If $\gamma\in\Pi(\mu,\nu)$, then $\gamma\circ s\in\Pi(\nu,\mu)$ and $I(\gamma\circ s)=I(\gamma)$.

6. (Weak-star closedness)¶ Let $(\mu_{m})$ and $(\nu_{m})$ be sequences in $\Sigma_{d,R}$ converging weak-star to $\mu$ and $\nu$, and let $\gamma_{m}\in\Pi(\mu_{m},\nu_{m})$ for every $m\in\mathbb{N}$. Then there are a strictly increasing sequence $(m_{i})_{i\in\mathbb{N}}$ in $\mathbb{N}$ and $\gamma\in\Pi(\mu,\nu)$ such that $\gamma_{m_{i}}\to\gamma$ weak-star and the real sequence $(I(\gamma_{m_{i}}))_{i\in\mathbb{N}}$ converges to $I(\gamma)$.

7. (Displacement interpolants)¶ Let $\mu,\nu\in\Sigma_{d,R}$, $\gamma\in\Pi(\mu,\nu)$ and $t\in\mathbb{R}$ with $0\le t\le1$. Let $a^{t}$ be the $d$-tuple in $\mathcal{P}_{2d}$ with $a^{t}_{j}=(1-t)x_{j}+t\,x_{d+j}$, and put $\gamma_{t}=\gamma\circ\sigma_{a^{t}}$. Then $\gamma_{t}\in\Sigma_{d,R}$, $\gamma_{0}=\mu$ and $\gamma_{1}=\nu$. Moreover, $\gamma\circ\sigma_{(x_{1},\dots,x_{d},a^{t}_{1},\dots,a^{t}_{d})}\in\Pi(\mu,\gamma_{t})$ with cost $t^{2}I(\gamma)$, and $\gamma\circ\sigma_{(a^{t}_{1},\dots,a^{t}_{d},x_{d+1},\dots,x_{2d})}\in\Pi(\gamma_{t},\nu)$ with cost $(1-t)^{2}I(\gamma)$.