Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts

Let $n\in\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers, let $W_{n}$ be the set of words in the letters $1,\dots,n$, with empty word $\varnothing$, concatenation $uv$ and reversal $w^{\mathrm{rev}}$, and let $\mathcal{P}_{n}=\mathbb{C}\langle x_{1},\dots,x_{n}\rangle$ be the set of noncommutative polynomials, with the linear operations, monomials $x_{w}$, unit $1$ and variables $x_{j}$, product, adjoint $p\mapsto p^{*}$ and self-adjoint part $\mathcal{P}_{n,\mathrm{sa}}$ of that definition. Here $\mathbb{C}$ is the field of complex numbers with imaginary unit $i$ and conjugation $z\mapsto\overline{z}$, $\mathbb{R}\subseteq\mathbb{C}$ the real numbers, and sums over a finite index set are those of Sum over a Finite Index Set. Vector spaces and linear maps are as in Vector Space over a Field and Linear Map; $\mathbb{C}$ is regarded as a complex vector space over itself. For $a\in\mathcal{P}_{n}$ write $a^{2}=aa$, and let $\tfrac12\in\mathbb{R}$ be the multiplicative inverse of $1+1$.

1. (Vector space)¶ With its linear operations, $\mathcal{P}_{n}$ is a complex vector space whose zero vector is the zero polynomial.

2. (Linear extension from monomials)¶ (a) For every map $c:W_{n}\to\mathbb{C}$ there is exactly one linear map $\ell:\mathcal{P}_{n}\to\mathbb{C}$ with $\ell(x_{w})=c(w)$ for every $w\in W_{n}$; it is given by $\ell(0)=0$ and $\ell(p)=\sum_{w\in\operatorname{supp}p}p(w)\,c(w)$ for $p\neq0$. (b) For every $m\in\mathbb{N}$ and every map $c:W_{n}\to\mathcal{P}_{m}$ there is exactly one linear map $T:\mathcal{P}_{n}\to\mathcal{P}_{m}$ with $T(x_{w})=c(w)$ for every $w\in W_{n}$; it is given by $T(0)=0$ and, for $p\neq0$ and $v\in W_{m}$, $T(p)(v)=\sum_{w\in\operatorname{supp}p}p(w)\,c(w)(v)$.

3. (Monomials)¶ For all $u,v\in W_{n}$ and $p\in\mathcal{P}_{n}$: $x_{u}x_{v}=x_{uv}$ and $1p=p1=p$.

4. (Algebra)¶ For all $p,q,r\in\mathcal{P}_{n}$ and $c\in\mathbb{C}$: $(pq)r=p(qr)$; $p(q+r)=pq+pr$ and $(p+q)r=pr+qr$; and $(cp)q=c(pq)=p(cq)$.

5. (Adjoint)¶ For all $p,q\in\mathcal{P}_{n}$, $c\in\mathbb{C}$ and $w\in W_{n}$: $(p+q)^{*}=p^{*}+q^{*}$, $(cp)^{*}=\overline{c}\,p^{*}$, $(pq)^{*}=q^{*}p^{*}$, $(p^{*})^{*}=p$, and $(x_{w})^{*}=x_{w^{\mathrm{rev}}}$; in particular $1^{*}=1$ and $x_{j}^{*}=x_{j}$ for every $j$ in the initial segment $[n]$.

6. (Self-adjoint part)¶ $\mathcal{P}_{n,\mathrm{sa}}$ contains $0$, $1$ and $x_{1},\dots,x_{n}$, and it is closed under sums and under multiplication by real numbers; with these operations it is a real vector space. For all $a,b\in\mathcal{P}_{n,\mathrm{sa}}$ and $p\in\mathcal{P}_{n}$, the polynomials $ab+ba$, $i(ab-ba)$, $p+p^{*}$ and $p^{*}p$ are self-adjoint.

7. (Real and imaginary parts)¶ Every $p\in\mathcal{P}_{n}$ can be written in exactly one way as $p=a+ib$ with $a,b\in\mathcal{P}_{n,\mathrm{sa}}$, namely with $a=\tfrac12(p+p^{*})$ and $b=-\tfrac{i}{2}(p-p^{*})$; for this decomposition $p^{*}=a-ib$ and