The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $\lambda\in\Sigma_{d}$. For a word $w=(w_{1},\dots,w_{k})\in W_{d}$ of length $k\in\mathbb{N}$ and $l\in[k]$, $w_{<l}$ and $w_{>l}$ denote the words $(w_{1},\dots,w_{l-1})$ and $(w_{l+1},\dots,w_{k})$, the first being the empty word $\varnothing$ when $l=1$ and the second being $\varnothing$ when $l=k$. For $i,j\in[d]$, $\delta_{ij}$ is $1$ if $i=j$ and $0$ otherwise.

1. (Free difference quotient)¶ For $j\in[d]$, the free difference quotient of $\lambda$ in the variable $x_{j}$ is the linear map $\partial^{\lambda}_{j}:\mathcal{P}_{d}\to\mathbb{C}$ with $\partial^{\lambda}_{j}(1)=0$ and

$$\partial^{\lambda}_{j}(x_{w})=\sum_{l=1}^{k}\delta_{w_{l}j}\,\lambda(x_{w_{<l}})\,\lambda(x_{w_{>l}})$$

for every word $w\in W_{d}$ of length $k\in\mathbb{N}$; it exists and is unique by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension.

2. (Conjugate variables)¶ $\lambda$ has conjugate variables if there are $\xi_{1},\dots,\xi_{d}\in\mathcal{H}_{\lambda}$ with

$$\langle\xi_{j},\widehat{p}\rangle_{\mathcal{H}_{\lambda}}=\partial^{\lambda}_{j}(p)\qquad\text{for every }j\in[d]\text{ and every }p\in\mathcal{P}_{d}.$$

Such vectors are unique: for two choices of $\xi_{j}$ the difference is orthogonal to every class $\widehat{p}$, and the classes are dense in $\mathcal{H}_{\lambda}$ by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §dense. The $d$-tuple $\xi_{\lambda}=(\xi_{1},\dots,\xi_{d})\in\mathcal{H}_{\lambda}^{d}$ is then called the conjugate variables of $\lambda$.