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

Each result cited below is universally quantified over the data in its own statement. We work in the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation. Fix $n$, $H$ and $c$ as in the statement. 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 whose zero vector is the zero map $0$, and by Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §maps its sums and scalar multiples are formed pointwise: $(A+B)\zeta=A\zeta+B\zeta$ and $(aA)\zeta=a\,(A\zeta)$ for $A,B\in\mathcal{L}(H)$, $a\in\mathbb{C}$ and $\zeta\in H$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §vector-space, $\mathcal{P}_{n}$ is a complex vector space whose zero vector is the zero polynomial, with the pointwise operations of The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §linear. By Complex Hilbert Space, $H$ is a complex inner product space, and its inner product is additive and homogeneous in the second argument by conditions 2 and 3 of Complex Inner Product Space.

Step 1 (operators are determined by their pairings). Let $A,B\in\mathcal{L}(H)$ satisfy $\langle\eta,A\zeta\rangle=\langle\eta,B\zeta\rangle$ for all $\eta,\zeta\in H$. Fix $\zeta\in H$ and put $u=A\zeta+(-1)B\zeta$. By conditions 2 and 3 of Complex Inner Product Space and the hypothesis with $\eta=u$,

so $u=0_{H}$ by claim 4 of Elementary Properties of a Complex Inner Product. By claim 5 of Elementary Identities in a Vector Space, $(-1)B\zeta=-B\zeta$, so $(-B\zeta)+A\zeta=0_{H}$ (addition is commutative), and also $(-B\zeta)+B\zeta=0_{H}$. By the uniqueness of additive inverses (claim 2 of Elementary Identities in a Vector Space, applied to the vector $-B\zeta$) we get $A\zeta=B\zeta$. As $\zeta$ was arbitrary, the maps $A$ and $B$ coincide: $A=B$.

Step 2 (enumerated sums and their pairings). Let $p\in\mathcal{P}_{n}$ with $p\neq0$. Since $p$ is not the zero polynomial, $p(w)\neq0$ for some $w\in W_{n}$, so $\operatorname{supp}p$ is nonempty; it is finite by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §polynomials. By Finite Set it has $N$ elements for some $N\in\mathbb{N}$, this $N$ is unique by Number of Elements of a Set, and by that definition there is at least one bijection $[N]\to\operatorname{supp}p$. For every such bijection $\phi$ put

a finite sum in the vector space $\mathcal{L}(H)$. We claim that for all $\eta,\zeta\in H$

Fix $\eta,\zeta\in H$. The map $E_{\zeta}:\mathcal{L}(H)\to H$, $E_{\zeta}(A)=A\zeta$, is linear because the operations of $\mathcal{L}(H)$ are pointwise. Claim 4 of Properties of Finite Sums of Vectors, applied to $E_{\zeta}$, together with $E_{\zeta}\bigl(p(\phi(k))\,c(\phi(k))\bigr)=p(\phi(k))\,\bigl(c(\phi(k))\zeta\bigr)$, gives

a finite sum in $H$. Claim 6 of Properties of Finite Sums of Vectors, in the complex inner product space $H$ with coefficients $p(\phi(k))$, then gives

where $f:\operatorname{supp}p\to\mathbb{C}$ is the map $f(w)=p(w)\,\langle\eta,c(w)\zeta\rangle$. Since $\operatorname{supp}p$ has $N$ elements and $\phi$ is a bijection, the last sum is $\sum_{w\in\operatorname{supp}p}f(w)$ by Sum over a Finite Index Set. This proves (A). The right side of (A) does not involve $\phi$; hence, if $\phi,\psi:[N]\to\operatorname{supp}p$ are two bijections, then $\langle\eta,S_{\phi}\zeta\rangle=\langle\eta,S_{\psi}\zeta\rangle$ for all $\eta,\zeta\in H$, and $S_{\phi}=S_{\psi}$ by Step 1.

Step 3 (existence). Define $\ell:\mathcal{P}_{n}\to\mathcal{L}(H)$ by $\ell(0)=0$ and, for $p\neq0$, $\ell(p)=S_{\phi}$ with $N$, $\phi$ as in Step 2; by Step 2 this does not depend on the choice of the bijection $\phi$, so $\ell$ is well defined. We first show: for every $p\in\mathcal{P}_{n}$, every nonempty finite set $F\subseteq W_{n}$ with $\operatorname{supp}p\subseteq F$, and all $\eta,\zeta\in H$,

If $p\neq0$, then by (A) the left side is the sum over $\operatorname{supp}p$; every $w\in F\setminus\operatorname{supp}p$ has $p(w)=0$, so its term is $0$, and the second part of Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing (with $G=\operatorname{supp}p$) gives (B). If $p=0$, then $\ell(0)\zeta=0\zeta=0_{H}$ because $\ell(0)$ is the zero map, so the left side is $0$ by claim 3 of Elementary Properties of a Complex Inner Product; every term on the right is $0\cdot\langle\eta,c(w)\zeta\rangle=0$, so the right side is $0$ by the first part of Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing.

Linearity. Let $p,q\in\mathcal{P}_{n}$ and $a\in\mathbb{C}$, and let $F=(\operatorname{supp}p\cup\operatorname{supp}q)\cup\{\varnothing\}$. The union $\operatorname{supp}p\cup\operatorname{supp}q$ is finite by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §linear, so $F$ is finite by claim 1 of Peeling an Element off a Finite Set, and Unions of Finite Sets, and $F$ is nonempty. By The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §linear, $\operatorname{supp}(p+q)\subseteq\operatorname{supp}p\cup\operatorname{supp}q\subseteq F$ and $\operatorname{supp}(ap)\subseteq\operatorname{supp}p\subseteq F$, and $(p+q)(w)=p(w)+q(w)$, $(ap)(w)=a\,p(w)$. Fix $\eta,\zeta\in H$ and write $g(w)=\langle\eta,c(w)\zeta\rangle$. Applying (B) to $p+q$, $p$ and $q$ with this $F$, distributivity in $\mathbb{C}$ and claim 3 of Properties of a Sum over a Finite Index Set,

and the last expression equals $\langle\eta,(\ell(p)+\ell(q))\zeta\rangle$ because sums in $\mathcal{L}(H)$ are pointwise and by condition 2 of Complex Inner Product Space. Likewise, applying (B) to $ap$ and $p$, associativity of multiplication in $\mathbb{C}$ and claim 4 of Properties of a Sum over a Finite Index Set,

which equals $\langle\eta,(a\,\ell(p))\zeta\rangle$ because scalar multiples in $\mathcal{L}(H)$ are pointwise and by condition 3 of Complex Inner Product Space. As $\eta,\zeta$ were arbitrary, Step 1 gives $\ell(p+q)=\ell(p)+\ell(q)$ and $\ell(ap)=a\,\ell(p)$; thus $\ell$ is linear in the sense of Linear Map.

Values on monomials. Let $w\in W_{n}$. By The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials, $x_{w}(w)=1\neq0$, so $x_{w}\neq0$, and $\operatorname{supp}x_{w}=\{w\}$. Since $[1]=\{1\}$, the map $\phi:[1]\to\{w\}$ with $\phi(1)=w$ is a bijection, so $\operatorname{supp}x_{w}$ has $1$ element, and by the first identity of claim 1 of Properties of Finite Sums of Vectors,

the last equality by the unit axiom of Vector Space over a Field. So $\ell$ is a linear map with $\ell(x_{w})=c(w)$ for every $w\in W_{n}$.

Step 4 (uniqueness). Let $\ell':\mathcal{P}_{n}\to\mathcal{L}(H)$ be linear with $\ell'(x_{w})=c(w)$ for every $w\in W_{n}$. First, $\ell'(0)=\ell'(0\cdot0)=0\cdot\ell'(0)=0=\ell(0)$, using claim 3 of Elementary Identities in a Vector Space in $\mathcal{P}_{n}$ and in $\mathcal{L}(H)$. Now let $p\neq0$, with $N$ and a bijection $\phi:[N]\to\operatorname{supp}p$ as in Step 2, and let $r=\sum_{k=1}^{N}p(\phi(k))\,x_{\phi(k)}$, a finite sum in the vector space $\mathcal{P}_{n}$. We show $r=p$. Fix $v\in W_{n}$. The map $\mathcal{P}_{n}\to\mathbb{C}$, $s\mapsto s(v)$, is linear ($\mathbb{C}$ being a vector space over itself), since the operations of The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §linear are pointwise; so by claim 4 of Properties of Finite Sums of Vectors

If $v\in\operatorname{supp}p$, surjectivity of $\phi$ gives $i\in[N]$ with $\phi(i)=v$; for $k\neq i$ injectivity gives $\phi(k)\neq v$, hence $x_{\phi(k)}(v)=0$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials and the $k$-th term is $0$; so claim 7 of Properties of Finite Sums of Vectors gives $r(v)=p(v)\,x_{v}(v)=p(v)$. If $v\notin\operatorname{supp}p$, then $p(v)=0$, and every $\phi(k)$ lies in $\operatorname{supp}p$, so $\phi(k)\neq v$ and every term is $0$; the last sentence of claim 7 of Properties of Finite Sums of Vectors gives $r(v)=0=p(v)$. Hence $r=p$ as maps $W_{n}\to\mathbb{C}$. By claim 4 of Properties of Finite Sums of Vectors for the linear map $\ell'$, the homogeneity condition of Linear Map and the hypothesis on $\ell'$,

Thus $\ell'=\ell$, and Clause 1 holds.

Step 5 (Clause 2). By Step 4, the map of Clause 1 is the map $\ell$ of Step 3. By construction $\ell(0)=0$. For $p\neq0$, $N$ the number of elements of $\operatorname{supp}p$ and any bijection $\phi:[N]\to\operatorname{supp}p$, we have $\ell(p)=S_{\phi}$ by the definition of $\ell$ and Step 2, which is the first formula of Clause 2, and the second formula is (A).