Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $m,n,r\in\mathbb{N}$, let $T=(A,c)$ be an affine datum from $m$ to $n$ variables and $S$ one from $n$ to $r$ variables, with tuples $a^{T}$, $a^{S}$ and affine substitutions $\sigma_{T}$, $\sigma_{S}$; let $S\circ T$ be their composite and $\mathrm{id}_{m}$ the identity datum. For a tracial state $\lambda$ on $\mathcal{P}_{m}$ and $i,j\in[m]$ write

the first and quadratic moments of $\lambda$, so that $M(\lambda)=\sum_{i=1}^{m}\mathrm{m}_{ii}(\lambda)$.

1. (Self-adjointness)¶ $a^{T}$ is an $n$-tuple in $\mathcal{P}_{m,\mathrm{sa}}$; for every tracial state $\lambda$ on $\mathcal{P}_{m}$, $\lambda\circ\sigma_{T}$ is a tracial state on $\mathcal{P}_{n}$; and $\lambda\circ\sigma_{T}\in\Sigma_{n}$ for every $\lambda\in\Sigma_{m}$.

2. (Composition)¶ $\sigma_{T}\circ\sigma_{S}=\sigma_{S\circ T}$ as maps from $\mathcal{P}_{r}$ to $\mathcal{P}_{m}$, and $\sigma_{\mathrm{id}_{m}}$ is the identity map of $\mathcal{P}_{m}$. Consequently $(\lambda\circ\sigma_{T})\circ\sigma_{S}=\lambda\circ\sigma_{S\circ T}$ and $\lambda\circ\sigma_{\mathrm{id}_{m}}=\lambda$ for every tracial state $\lambda$ on $\mathcal{P}_{m}$.

3. (Moments)¶ Let $\lambda$ be a tracial state on $\mathcal{P}_{m}$. For all $i,j\in[m]$ and all real $\xi_{1},\dots,\xi_{m}$, the numbers $\mathrm{m}_{i}(\lambda)$ and $\mathrm{m}_{ij}(\lambda)$ are real, and

For all $i,k\in[n]$,

4. (Coordinate data)¶ Let $d\in\mathbb{N}$, and let $\mathrm{pr}^{1}$, $\mathrm{pr}^{2}$, $D$ and $\mathrm{diag}$ be the coordinate data. Then $\sigma_{\mathrm{pr}^{1}}=\iota^{1}$ and $\sigma_{\mathrm{pr}^{2}}=\iota^{2}$ are the marginal substitutions; $\sigma_{\mathrm{diag}}$ is the substitution of the $2d$-tuple $(x_{1},\dots,x_{d},x_{1},\dots,x_{d})$ in $\mathcal{P}_{d}$; $\mathrm{pr}^{1}\circ\mathrm{diag}=\mathrm{pr}^{2}\circ\mathrm{diag}=\mathrm{id}_{d}$; and $D\circ\mathrm{diag}$ is the affine datum from $d$ to $d$ variables both of whose maps have value $0$. ¶For every tracial state $\gamma$ on $\mathcal{P}_{2d}$,