Linear Maps from Noncommutative Polynomials into Bounded Operators: Existence and Uniqueness from Values on Monomials

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 $c:W_{n}\to\mathcal{L}(H)$ be a map. By Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations, $\mathcal{L}(H)$ is a complex vector space with zero vector the zero map $0$; sums $\sum_{k=1}^{N}$ of elements of $\mathcal{L}(H)$ are the finite sums in it, and sums $\sum_{w\in F}$ of complex numbers over a nonempty finite set $F$ are those of Sum over a Finite Index Set. For $p\in\mathcal{P}_{n}$, $\operatorname{supp}p$ is its support.

1. (Extension)¶ There is exactly one linear map $\ell:\mathcal{P}_{n}\to\mathcal{L}(H)$ with $\ell(x_{w})=c(w)$ for every $w\in W_{n}$.

2. (Formula)¶ This map satisfies $\ell(0)=0$. Let $p\in\mathcal{P}_{n}$ with $p\neq0$, let $N$ be the number of elements of the nonempty finite set $\operatorname{supp}p$, and let $\phi:[N]\to\operatorname{supp}p$ be a bijection. Then