Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition

Let $l,m,n\in\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers, let $W_{n}$ be the set of words in the letters $1,\dots,n$ with concatenation $uv$, and for $r\in\mathbb{N}$ let $\mathcal{P}_{r}=\mathbb{C}\langle x_{1},\dots,x_{r}\rangle$ be the noncommutative polynomials in $r$ variables, with product, unit $1$, monomials $x_{w}$, variables $x_{j}$, adjoint $p\mapsto p^{*}$ and self-adjoint part $\mathcal{P}_{r,\mathrm{sa}}$. Let $a=(a_{1},\dots,a_{n})$ be an $n$-tuple in $\mathcal{P}_{m}$, with products along words $a_{w}$ and substitution $\sigma_{a}:\mathcal{P}_{n}\to\mathcal{P}_{m}$. Linear maps are as in Linear Map, and $[n]$ is the initial segment determined by $n$.

1. (Values)¶ $a_{uv}=a_{u}a_{v}$ for all $u,v\in W_{n}$; $\sigma_{a}(x_{w})=a_{w}$ for every $w\in W_{n}$, $\sigma_{a}(1)=1$, and $\sigma_{a}(x_{j})=a_{j}$ for every $j\in[n]$.

2. (Homomorphism)¶ $\sigma_{a}(pq)=\sigma_{a}(p)\,\sigma_{a}(q)$ for all $p,q\in\mathcal{P}_{n}$.

3. (Uniqueness)¶ If $T:\mathcal{P}_{n}\to\mathcal{P}_{m}$ is a linear map with $T(pq)=T(p)T(q)$ for all $p,q\in\mathcal{P}_{n}$, $T(1)=1$, and $T(x_{j})=a_{j}$ for every $j\in[n]$, then $T=\sigma_{a}$.

4. (Adjoints)¶ If $a_{j}\in\mathcal{P}_{m,\mathrm{sa}}$ for every $j\in[n]$, then $\sigma_{a}(p^{*})=\sigma_{a}(p)^{*}$ for every $p\in\mathcal{P}_{n}$, and $\sigma_{a}$ maps $\mathcal{P}_{n,\mathrm{sa}}$ into $\mathcal{P}_{m,\mathrm{sa}}$.

5. (Composition)¶ Let $b=(b_{1},\dots,b_{m})$ be an $m$-tuple in $\mathcal{P}_{l}$, and let $c$ be the $n$-tuple in $\mathcal{P}_{l}$ with $c_{j}=\sigma_{b}(a_{j})$. Then $\sigma_{b}\bigl(\sigma_{a}(p)\bigr)=\sigma_{c}(p)$ for every $p\in\mathcal{P}_{n}$.

6. (Identity)¶ For the $n$-tuple $x=(x_{1},\dots,x_{n})$ of variables in $\mathcal{P}_{n}$, $\sigma_{x}(p)=p$ for every $p\in\mathcal{P}_{n}$.