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

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.

Throughout, $T_{j}$ is the $j$-th component of $T$ and $T_{(j)}$ is the product along the letter $(j)$; the same notation is used for the other tuples below. Since $S$ also names a tuple in clause 4, we write $k+1$ for the successor $S(k)$ of $k\in\mathbb{N}$, as permitted by claim 1 of Natural Numbers. 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, and composition in it, being composition of maps, is associative, distributes over sums, commutes with scalar multiples and satisfies $IA=AI=A$. Clauses 1 and 2 are proved for an arbitrary complex Hilbert space and an arbitrary $n$-tuple of bounded operators on it; the later clauses apply them to other tuples and spaces.

Uniqueness principle. If $\ell_{1},\ell_{2}:\mathcal{P}_{n}\to\mathcal{L}(H)$ are linear maps with $\ell_{1}(x_{w})=\ell_{2}(x_{w})$ for every $w\in W_{n}$, then $\ell_{1}=\ell_{2}$, since both are linear maps with the prescribed values $c(w)=\ell_{1}(x_{w})$ and such a map is unique by Linear Maps from Noncommutative Polynomials into Bounded Operators: Existence and Uniqueness from Values on Monomials §extension. In particular, by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation, a linear map $\ell:\mathcal{P}_{n}\to\mathcal{L}(H)$ with $\ell(x_{w})=T_{w}$ for every $w\in W_{n}$ is the evaluation $p\mapsto p(T)$.

Clause 1. By The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials, Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation and Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products, $1(T)=x_{\varnothing}(T)=T_{\varnothing}=I$. For $j\in[n]$ the letter $(j)$ is the word of length $1$ with value $j$ at $1$, so $T_{(j)}=\pi(1)=T_{j}$ by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products, and $x_{j}(T)=x_{(j)}(T)=T_{(j)}=T_{j}$.

We first show

If $u=\varnothing$, then $u(j)=(j)$ by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation, and $T_{(j)}=I\,T_{(j)}$. Let $u$ have length $k$. By Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation, $u(j)$ has length $k+1$, its restriction to $[k]$ is $u$, and its value at $k+1$ is $j$. Let $\pi:[k+1]\to\mathcal{L}(H)$ be the map of Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products for the word $u(j)$. As $[k]\subseteq[k+1]$ by claim 3 of Basic Properties of Initial Segments of the Natural Numbers, the restriction of $\pi$ to $[k]$ satisfies the two conditions that define the map for $u$, so it is that map by the uniqueness in Existence and Uniqueness of Iterates of a Binary Operation. Hence $\pi(k)=T_{u}$ and $T_{u(j)}=\pi(k+1)=\pi(k)\,T_{j}=T_{u}\,T_{(j)}$.

Now let $u,v\in W_{n}$. If $u=\varnothing$ or $v=\varnothing$, then $uv$ is $v$ or $u$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid, and $T_{uv}=T_{u}T_{v}$ because $T_{\varnothing}=I$. For nonempty $v$ we argue by induction on the length of $v$, using Principle of Induction for the Natural Numbers for the inductive set $E_{1}$ of those $k\in\mathbb{N}$ such that $T_{uv}=T_{u}T_{v}$ for every $u\in W_{n}$ and every word $v$ of length $k$. A word of length $1$ is a letter by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, so $1\in E_{1}$ by (R). Let $k\in E_{1}$ and let $v$ have length $k+1$. By Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, $v=v'(j)$ with $v'$ of length $k$, and $uv=(uv')(j)$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid. By (R), $k\in E_{1}$, associativity of composition and (R) again,

so $k+1\in E_{1}$. Hence $E_{1}=\mathbb{N}$; as every nonempty word has a length, $T_{uv}=T_{u}T_{v}$ for all $u,v\in W_{n}$.

Clause 2. Fix $v\in W_{n}$ and let $\ell_{1}(p)=(px_{v})(T)$ and $\ell_{2}(p)=p(T)\,T_{v}$ for $p\in\mathcal{P}_{n}$. Both maps are linear: $p\mapsto px_{v}$ is linear by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, evaluation is linear, and $A\mapsto AT_{v}$ is linear because composition distributes over sums and commutes with scalar multiples. For $u\in W_{n}$, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials and clause 1 give $\ell_{1}(x_{u})=x_{uv}(T)=T_{uv}=T_{u}T_{v}=\ell_{2}(x_{u})$. By the uniqueness principle,

Now fix $p\in\mathcal{P}_{n}$ and let $m_{1}(q)=(pq)(T)$ and $m_{2}(q)=p(T)\,q(T)$ for $q\in\mathcal{P}_{n}$. Both are linear, by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, the linearity of evaluation and the linearity of $A\mapsto p(T)A$; and $m_{1}(x_{v})=(px_{v})(T)=p(T)\,T_{v}=m_{2}(x_{v})$ for every $v\in W_{n}$. The uniqueness principle gives $m_{1}=m_{2}$, that is, $(pq)(T)=p(T)\,q(T)$.

Clause 3. Assume that every $T_{j}$ is self-adjoint. As $H$ is a Hilbert space, every $A\in\mathcal{L}(H)$ has an adjoint $A^{*}\in\mathcal{L}(H)$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint, and adjoints are unique by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-unique; so each rule of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus used below is an equality between adjoints. By that clause each $T_{j}$ is an adjoint of itself, so $T_{j}^{*}=T_{j}$.

We first show $(T_{w})^{*}=T_{w^{\mathrm{rev}}}$ for every $w\in W_{n}$. For $w=\varnothing$ this reads $I^{*}=I$, which holds by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, since $\varnothing^{\mathrm{rev}}=\varnothing$ by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §reversal. For nonempty words we argue by induction on the length, using Principle of Induction for the Natural Numbers for the inductive set $E_{3}$ of those $k\in\mathbb{N}$ for which the identity holds for every word of length $k$. For a letter, $(j)^{\mathrm{rev}}=(j)$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §reversal and $T_{(j)}^{*}=T_{j}^{*}=T_{j}=T_{(j)}$, so $1\in E_{3}$. Let $k\in E_{3}$ and let $w$ have length $k+1$; write $w=w'(j)$ with $w'$ of length $k$, by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter. By clause 1, the rule $(RA)^{*}=A^{*}R^{*}$ of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, $k\in E_{3}$, $1\in E_{3}$, clause 1 again and Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §reversal,

so $k+1\in E_{3}$. Hence $E_{3}=\mathbb{N}$, and the identity holds for every word.

Next let $\ell(p)=\bigl(p^{*}(T)\bigr)^{*}$ for $p\in\mathcal{P}_{n}$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, the linearity of evaluation and Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, for $p,q\in\mathcal{P}_{n}$ and $c\in\mathbb{C}$,

so $\ell$ is linear. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, the identity just proved and Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §reversal, $\ell(x_{w})=\bigl(x_{w^{\mathrm{rev}}}(T)\bigr)^{*}=(T_{w^{\mathrm{rev}}})^{*}=T_{(w^{\mathrm{rev}})^{\mathrm{rev}}}=T_{w}$ for every $w\in W_{n}$. By the uniqueness principle, $\bigl(p^{*}(T)\bigr)^{*}=p(T)$ for every $p\in\mathcal{P}_{n}$. By Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, $p^{*}(T)$ is the adjoint of $\bigl(p^{*}(T)\bigr)^{*}=p(T)$, that is, $p^{*}(T)=p(T)^{*}$. Finally, let $a\in\mathcal{P}_{n,\mathrm{sa}}$. Then $a^{*}=a$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §self-adjoint, so $a(T)=a^{*}(T)=a(T)^{*}$; thus $a(T)$ is an adjoint of itself, and it is self-adjoint by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus.

Clause 4. Let $\ell_{1}(p)=\bigl(\sigma_{a}(p)\bigr)(S)$ and $\ell_{2}(p)=p\bigl(a(S)\bigr)$ for $p\in\mathcal{P}_{n}$. The map $\ell_{1}$ is linear as the composite of the linear map $\sigma_{a}$ of Substitution of Noncommutative Polynomials into the Variables §substitution with evaluation at $S$, and $\ell_{2}$ is evaluation at $a(S)$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $\ell_{1}(x_{w})=(a_{w})(S)$, while $\ell_{2}(x_{w})=(a(S))_{w}$. By the uniqueness principle it suffices to show $(a_{w})(S)=(a(S))_{w}$ for every $w\in W_{n}$. For $w=\varnothing$, $a_{\varnothing}=1$ by Substitution of Noncommutative Polynomials into the Variables §word-products, and $1(S)=I=(a(S))_{\varnothing}$ by clause 1 and Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products. For nonempty words we argue by induction on the length, using Principle of Induction for the Natural Numbers for the inductive set $E_{4}$ of those $k\in\mathbb{N}$ for which the identity holds for every word of length $k$. For a letter, $a_{(j)}=\sigma_{a}(x_{j})=a_{j}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, and $(a(S))_{(j)}=a_{j}(S)$ by clause 1 for the tuple $a(S)$; so $1\in E_{4}$. Let $k\in E_{4}$ and let $w=w'(j)$ have length $k+1$, with $w'$ of length $k$ (Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter). By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, clause 2 for $S$, $k\in E_{4}$, $1\in E_{4}$ and clause 1 for $a(S)$,

so $k+1\in E_{4}$. Hence $E_{4}=\mathbb{N}$, and $\ell_{1}=\ell_{2}$.

Clause 5. Write $\Phi(T)$ for the $n$-tuple $(\Phi(T_{1}),\dots,\Phi(T_{n}))$ in $\mathcal{L}(K)$; clause 1 applies to it with $K$ in place of $H$. We first show that $T_{w}\in\mathcal{A}$ and $\Phi(T_{w})=(\Phi(T))_{w}$ for every $w\in W_{n}$. For $w=\varnothing$, $T_{\varnothing}=I\in\mathcal{A}$ and $\Phi(I)=I=(\Phi(T))_{\varnothing}$. For nonempty words we argue by induction on the length, using Principle of Induction for the Natural Numbers for the inductive set $E_{5}$ of those $k\in\mathbb{N}$ for which both statements hold for every word of length $k$. For a letter, $T_{(j)}=T_{j}\in\mathcal{A}$ and $\Phi(T_{(j)})=\Phi(T_{j})=(\Phi(T))_{(j)}$ by clause 1 for $T$ and for $\Phi(T)$, so $1\in E_{5}$. Let $k\in E_{5}$ and let $w=w'(j)$ have length $k+1$, with $w'$ of length $k$ (Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter). Then $T_{w}=T_{w'}T_{(j)}\in\mathcal{A}$ by clause 1 and closure under composition, and $\Phi(T_{w})=\Phi(T_{w'})\,\Phi(T_{(j)})=(\Phi(T))_{w'}\,(\Phi(T))_{(j)}=(\Phi(T))_{w}$ by the multiplicativity of $\Phi$, $k\in E_{5}$, $1\in E_{5}$ and clause 1 for $\Phi(T)$. So $k+1\in E_{5}$, and $E_{5}=\mathbb{N}$.

Next, sums in $\mathcal{L}(H)$ and $\mathcal{L}(K)$ are the finite sums of Finite Sum Notation in a Vector Space. We show by induction on $N$, using Principle of Induction for the Natural Numbers for the inductive set $F_{5}$ of those $N\in\mathbb{N}$ such that for every map $v:[N]\to\mathcal{A}$ the sum $\sum_{k=1}^{N}v_{k}$ lies in $\mathcal{A}$ and $\Phi\bigl(\sum_{k=1}^{N}v_{k}\bigr)=\sum_{k=1}^{N}\Phi(v_{k})$, that $F_{5}=\mathbb{N}$. By claim 1 of Properties of Finite Sums of Vectors the two sums for $N=1$ are $v_{1}$ and $\Phi(v_{1})$, so $1\in F_{5}$. Let $N\in F_{5}$, let $v:[N+1]\to\mathcal{A}$, and let $v'$ be the restriction of $v$ to $[N]$, which is contained in $[N+1]$ and contains $N$ by claims 1 and 3 of Basic Properties of Initial Segments of the Natural Numbers. By claim 1 of Properties of Finite Sums of Vectors, applied in $\mathcal{L}(H)$ and in $\mathcal{L}(K)$,

Since $N\in F_{5}$, closure of $\mathcal{A}$ under sums and additivity of $\Phi$ give $N+1\in F_{5}$. Hence $F_{5}=\mathbb{N}$.

Now let $p\in\mathcal{P}_{n}$. If $p=0$, then $p(T)=0$ and $p(\Phi(T))=0$ by Linear Maps from Noncommutative Polynomials into Bounded Operators: Existence and Uniqueness from Values on Monomials §formula, $0=0\,I\in\mathcal{A}$, and $\Phi(0)=\Phi(0\,I)=0\,\Phi(I)=0$. If $p\neq0$, let $N$ and $\phi$ be as in Linear Maps from Noncommutative Polynomials into Bounded Operators: Existence and Uniqueness from Values on Monomials §formula. That clause, applied to evaluation at $T$ (the linear map with values $c(w)=T_{w}$) and to evaluation at $\Phi(T)$ (with $c(w)=(\Phi(T))_{w}$, on $K$), gives

The terms $v_{k}=p(\phi(k))\,T_{\phi(k)}$ lie in $\mathcal{A}$ by the first step and closure under multiplication by complex numbers, and $\Phi(v_{k})=p(\phi(k))\,\Phi(T_{\phi(k)})=p(\phi(k))\,(\Phi(T))_{\phi(k)}$ by homogeneity of $\Phi$ and the first step. Since $N\in F_{5}$, we get $p(T)\in\mathcal{A}$ and $\Phi(p(T))=\sum_{k=1}^{N}\Phi(v_{k})=p(\Phi(T))$.

Clause 6. Since $\lambda\in\Sigma_{n}$, there is a real $r>0$ with $\lambda\in\Sigma_{n,r}$, by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law; so Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation applies with $d=n$. The space $\mathcal{H}_{\lambda}$ is a complex Hilbert space by The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns, and $L_{q}\in\mathcal{L}(\mathcal{H}_{\lambda})$ for every $q\in\mathcal{P}_{n}$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §multiplication. Write $L_{x}$ for the $n$-tuple $(L_{x_{1}},\dots,L_{x_{n}})$. The maps $p\mapsto p(L_{x})$ and $p\mapsto L_{p}$ from $\mathcal{P}_{n}$ to $\mathcal{L}(\mathcal{H}_{\lambda})$ are linear, the second by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §algebra. By the uniqueness principle on $\mathcal{H}_{\lambda}$ it suffices to show $L_{x_{w}}=(L_{x})_{w}$ for every $w\in W_{n}$. For $w=\varnothing$, $L_{x_{\varnothing}}=L_{1}=I=(L_{x})_{\varnothing}$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §algebra. For nonempty words we argue by induction on the length, using Principle of Induction for the Natural Numbers for the inductive set $E_{6}$ of those $k\in\mathbb{N}$ for which the identity holds for every word of length $k$. For a letter, $L_{x_{(j)}}=L_{x_{j}}=(L_{x})_{(j)}$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials and clause 1 for $L_{x}$, so $1\in E_{6}$. Let $k\in E_{6}$ and let $w=w'(j)$ have length $k+1$, with $w'$ of length $k$ (Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter). Then $x_{w}=x_{w'}x_{(j)}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, and by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §algebra, $k\in E_{6}$, $1\in E_{6}$ and clause 1 for $L_{x}$,

so $k+1\in E_{6}$. Hence $E_{6}=\mathbb{N}$, and $p(L_{x_{1}},\dots,L_{x_{n}})=L_{p}$ for every $p\in\mathcal{P}_{n}$.

Clause 7. Every operator norm is nonnegative: $\lVert A\rVert_{\mathrm{op}}$ is a bound for $A$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound, and bounds are nonnegative by Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded. We argue by induction on $k$, using Principle of Induction for the Natural Numbers for the inductive set $E_{7}$ of those $k\in\mathbb{N}$ with $\lVert T_{w}\rVert_{\mathrm{op}}\le R^{k}$ for every word $w$ of length $k$. A word of length $1$ is a letter $(j)$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, and $\lVert T_{(j)}\rVert_{\mathrm{op}}=\lVert T_{j}\rVert_{\mathrm{op}}\le R=R^{1}$ by clause 1 and claim 1 of Properties of Natural Number Powers in a Field; so $1\in E_{7}$. Let $k\in E_{7}$ and let $w$ have length $k+1$; write $w=w'(j)$ with $w'$ of length $k$, by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter. By clause 1 and Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations,

The second inequality is claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative factor $\lVert T_{w'}\rVert_{\mathrm{op}}$; the third is the same claim with the nonnegative factor $R$, applied to $\lVert T_{w'}\rVert_{\mathrm{op}}\le R^{k}$ (which holds as $k\in E_{7}$), together with commutativity of multiplication; the final equality is claim 1 of Properties of Natural Number Powers in a Field. Hence $k+1\in E_{7}$, so $E_{7}=\mathbb{N}$, which is clause 7.