Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $n\in\mathbb{N}$, let $H$ be a complex Hilbert space and let $T=(T_{1},\dots,T_{n})$ be an $n$-tuple in $\mathcal{L}(H)$. For $w\in W_{n}$ and $p\in\mathcal{P}_{n}$, $T_{w}$ and $p(T)$ are the product along $w$ and the value of $p$ at $T$; for $p\in\mathcal{P}_{n}$, $p^{*}(T)$ stands for $(p^{*})(T)$.

1. (Values)¶ $T_{uv}=T_{u}T_{v}$ for all $u,v\in W_{n}$; $1(T)=I$, and $x_{j}(T)=T_{j}$ for every $j\in[n]$.

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

3. (Adjoints)¶ If $T_{j}$ is self-adjoint for every $j\in[n]$, then $p^{*}(T)=p(T)^{*}$ for every $p\in\mathcal{P}_{n}$; in particular $a(T)$ is self-adjoint for every $a\in\mathcal{P}_{n,\mathrm{sa}}$.

4. (Substitution)¶ Let $m\in\mathbb{N}$, let $a=(a_{1},\dots,a_{n})$ be an $n$-tuple in $\mathcal{P}_{m}$, let $S$ be an $m$-tuple in $\mathcal{L}(H)$, and let $a(S)=(a_{1}(S),\dots,a_{n}(S))$, an $n$-tuple in $\mathcal{L}(H)$. Then $\bigl(\sigma_{a}(p)\bigr)(S)=p\bigl(a(S)\bigr)$ for every $p\in\mathcal{P}_{n}$.

5. (Transport)¶ Let $K$ be a complex Hilbert space, let $\mathcal{A}\subseteq\mathcal{L}(H)$ be a set that contains $I$ and $T_{1},\dots,T_{n}$ and is closed under sums, multiplication by complex numbers and composition, and let $\Phi:\mathcal{A}\to\mathcal{L}(K)$ be a map with

for all $A,B\in\mathcal{A}$ and $c\in\mathbb{C}$. Then, for every $p\in\mathcal{P}_{n}$, $p(T)\in\mathcal{A}$ and $\Phi\bigl(p(T)\bigr)=p\bigl(\Phi(T_{1}),\dots,\Phi(T_{n})\bigr)$, the value of $p$ at the $n$-tuple $(\Phi(T_{1}),\dots,\Phi(T_{n}))$ in $\mathcal{L}(K)$.

6. (GNS multiplication operators)¶ Let $\lambda\in\Sigma_{n}$, and let $L_{q}$ $(q\in\mathcal{P}_{n})$ be the left multiplication operators on the complex GNS space $\mathcal{H}_{\lambda}$ of $\lambda$ (this clause concerns the Hilbert space $\mathcal{H}_{\lambda}$ in place of $H$). Then $p(L_{x_{1}},\dots,L_{x_{n}})=L_{p}$ for every $p\in\mathcal{P}_{n}$, the left side being the value of $p$ at the $n$-tuple $(L_{x_{1}},\dots,L_{x_{n}})$ in $\mathcal{L}(\mathcal{H}_{\lambda})$.

7. (Bound)¶ Let $R>0$ be real with $\lVert T_{j}\rVert_{\mathrm{op}}\le R$ for every $j\in[n]$. Then $\lVert T_{w}\rVert_{\mathrm{op}}\le R^{k}$, the natural power, for every $k\in\mathbb{N}$ and every word $w\in W_{n}$ of length $k$.