Proof of Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation

Write $h=h_{\lambda}$, $h(u,v)=\lambda(u^{*}v)$, and $\beta=\operatorname{Re}h$, as in The Complex GNS Space of a Tracial State on Noncommutative Polynomials. Since $\lambda\in\Sigma_{d,r}$, $\lambda$ is a tracial state by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound; condition (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, $\lambda(ab)=\lambda(ba)$, is called traciality below.

Preliminaries. (P1) By The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns, $\mathcal{H}$ is the complex Hilbert completion $H_{h}$, a complex Hilbert space by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §hilbert, and $\widehat{u}=J_{h}(u)$ by The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes. By The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry, $u\mapsto\widehat{u}$ is complex-linear and $\langle\widehat{u},\widehat{v}\rangle=\lambda(u^{*}v)$ for all $u,v\in\mathcal{P}_{d}$; in particular $\lVert\widehat{u}\rVert^{2}=\lambda(u^{*}u)=\lVert u\rVert_{\lambda}^{2}$, so $\lVert\widehat{u}\rVert=\lVert u\rVert_{\lambda}$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root.

(P2) Let $M$ be a metric space and let $f,g:\mathcal{H}\to M$ be continuous with $f(\widehat{q})=g(\widehat{q})$ for every $q\in\mathcal{P}_{d}$. Then $f=g$. Indeed, let $\xi\in\mathcal{H}$. By The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §completion, $\mathcal{H}$ is the set $H_{\beta}$, so $\xi=[u]$ for a $\beta$-Cauchy sequence $u=(u_{k})_{k\in\mathbb{N}}$ in $\mathcal{P}_{d}$ by The Hilbert Completion of a Real Vector Space with a Positive Semidefinite Symmetric Bilinear Form §completion, and $(J_{\beta}u_{k})_{k\in\mathbb{N}}$ converges to $\xi$ in $H_{\beta}$ by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §dense. Here $J_{\beta}u_{k}=J_{h}u_{k}=\widehat{u_{k}}$ by The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §canonical-map, and the metric of $\mathcal{H}$ is that of $H_{\beta}$ because the norm of $\mathcal{H}$ is $|\cdot|$ by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §hilbert; so $\widehat{u_{k}}\to\xi$ in $\mathcal{H}$. By the definition of continuity at $\xi$ (given $\varepsilon>0$ take its $\delta$, and then $N$ with $\lVert\widehat{u_{k}}-\xi\rVert<\delta$ for $k\ge N$), $f(\widehat{u_{k}})\to f(\xi)$ and $g(\widehat{u_{k}})\to g(\xi)$ in $M$. These are the same sequence, so $f(\xi)=g(\xi)$ by Uniqueness of Limits in a Metric Space.

(P3) Every $A\in\mathcal{L}(\mathcal{H})$ is continuous: if $C$ is a bound for $A$, then $\lVert A\xi-A\eta\rVert=\lVert A(\xi-\eta)\rVert\le C\lVert\xi-\eta\rVert$ by linearity.

1. (Multiplication operators) Step 1: for every $p\in\mathcal{P}_{d}$ there is a real $C_{p}\ge0$ with $\lVert\widehat{pq}\rVert\le C_{p}\lVert\widehat{q}\rVert$ for all $q\in\mathcal{P}_{d}$. First let $p=x_{w}$ be a monomial; we argue by induction on the length of $w$. For the empty word, $x_{\varnothing}q=1q=q$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, and $C=1$ works. Otherwise $w$ is the concatenation of a one-letter word $(j)$ and a shorter word $w'$, so $x_{w}=x_{j}x_{w'}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials and $x_{w}q=x_{j}(x_{w'}q)$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra; by (P1), The Norm Bound of a Noncommutative Law: Multiplication by a Variable is Bounded, the Bound is Detected by Even Moments, and Laws Pull Back under Self-Adjoint Substitutions §multiplication (as $\lambda\in\Sigma_{d,r}$) and the induction hypothesis,

Now let $p$ be arbitrary; we argue by induction on the number of elements of the finite set $\operatorname{supp}p$. If it is empty, $p=0$ and $pq=(0p)q=0(pq)=0$ by claim 3 of Elementary Identities in a Vector Space in the complex vector space $\mathcal{P}_{d}$ (used twice) and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, so $C_{p}=0$ works. Otherwise pick $w\in\operatorname{supp}p$ and let $p'=p-p(w)x_{w}$. By The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §linear and The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials, $p'(v)=p(v)$ for $v\neq w$ and $p'(w)=0$, so $\operatorname{supp}p'=\operatorname{supp}p\setminus\{w\}$ has one element fewer. As $p=p'+p(w)x_{w}$, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra gives $pq=p'q+p(w)(x_{w}q)$, and by (P1) and claim 2 of The Induced Norm is a Norm, and Induces a Metric,

Step 2: $\lVert qp\rVert_{\lambda}=\lVert p^{*}q^{*}\rVert_{\lambda}$ and $\lVert q^{*}\rVert_{\lambda}=\lVert q\rVert_{\lambda}$ for all $p,q$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, $(qp)^{*}=p^{*}q^{*}$ and $(p^{*}q^{*})^{*}=qp$, so by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and traciality

and $\lambda((q^{*})^{*}q^{*})=\lambda(qq^{*})=\lambda(q^{*}q)$. Taking nonnegative square roots gives both identities. Hence, by (P1) and Step 1 for $p^{*}$, $\lVert\widehat{qp}\rVert=\lVert p^{*}q^{*}\rVert_{\lambda}\le C_{p^{*}}\lVert q^{*}\rVert_{\lambda}=C_{p^{*}}\lVert\widehat{q}\rVert$.

Step 3: existence and uniqueness. The maps $q\mapsto\widehat{pq}$ and $q\mapsto\widehat{qp}$ from $\mathcal{P}_{d}$ to $\mathcal{H}$ are complex-linear, by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and (P1), and by Steps 1 and 2 their values have squared norms at most $C_{p}^{2}\lVert\widehat{q}\rVert^{2}=C_{p}^{2}h(q,q)$ and $C_{p^{*}}^{2}h(q,q)$. By The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §extension-linear with $K=\mathcal{H}$, there are continuous $L_{p},R_{p}\in\mathcal{L}(\mathcal{H})$ with $L_{p}\widehat{q}=\widehat{pq}$ and $R_{p}\widehat{q}=\widehat{qp}$ for every $q$, and each is the only continuous map with its property; since elements of $\mathcal{L}(\mathcal{H})$ are continuous by (P3), each is the only element of $\mathcal{L}(\mathcal{H})$ with its property.

Step 4: the bounds for $x_{j}$. By (P1) and The Norm Bound of a Noncommutative Law: Multiplication by a Variable is Bounded, the Bound is Detected by Even Moments, and Laws Pull Back under Self-Adjoint Substitutions §multiplication, $\lVert\widehat{x_{j}q}\rVert=\lVert x_{j}q\rVert_{\lambda}\le r\lVert\widehat{q}\rVert$; and since $x_{j}^{*}=x_{j}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, Step 2 and the same claim give $\lVert\widehat{qx_{j}}\rVert=\lVert x_{j}q^{*}\rVert_{\lambda}\le r\lVert q^{*}\rVert_{\lambda}=r\lVert\widehat{q}\rVert$. So The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §extension-linear applies with $C=r$ and yields elements of $\mathcal{L}(\mathcal{H})$ with bound $r$ sending $\widehat{q}$ to $\widehat{x_{j}q}$, respectively $\widehat{qx_{j}}$; by the uniqueness in Step 3 these are $L_{x_{j}}$ and $R_{x_{j}}$, and $\lVert L_{x_{j}}\rVert_{\mathrm{op}}\le r$, $\lVert R_{x_{j}}\rVert_{\mathrm{op}}\le r$ 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.

2. (Algebra rules) By claim 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, every map occurring in the identities belongs to $\mathcal{L}(\mathcal{H})$. For $s\in\mathcal{P}_{d}$, by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials and the complex-linearity of $u\mapsto\widehat{u}$ from (P1),

So $L_{p}+L_{q}$, $cL_{p}$, $L_{p}L_{q}$ and $I$ send $\widehat{s}$ to $\widehat{(p+q)s}$, $\widehat{(cp)s}$, $\widehat{(pq)s}$ and $\widehat{1s}$, and $R_{p}+R_{q}$, $cR_{p}$, $R_{q}R_{p}$ and $I$ send $\widehat{s}$ to $\widehat{s(p+q)}$, $\widehat{s(cp)}$, $\widehat{s(pq)}$ and $\widehat{s1}$. By the uniqueness in claim 1 they equal $L_{p+q}$, $L_{cp}$, $L_{pq}$, $L_{1}$ and $R_{p+q}$, $R_{cp}$, $R_{pq}$, $R_{1}$.

3. (Adjoints) For $q,s\in\mathcal{P}_{d}$, by (P1), Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and traciality,

Let $(A,B)$ be $(L_{p^{*}},L_{p})$ or $(R_{p^{*}},R_{p})$. For fixed $x\in\mathcal{H}$ the maps $y\mapsto\langle x,y\rangle$ and $y\mapsto\langle y,x\rangle$ from $\mathcal{H}$ to $\mathbb{C}$, with the metric of claim 9 of Properties of Complex Conjugation and Modulus, are continuous: $|\langle x,y\rangle-\langle x,y'\rangle|=|\langle x,y-y'\rangle|\le\lVert x\rVert\,\lVert y-y'\rVert$ by additivity in the second argument and claim 1 of The Induced Norm is a Norm, and Induces a Metric, and likewise in the first argument using claim 1 of Elementary Properties of a Complex Inner Product. As $A$ and $B$ are continuous by (P3), for fixed $s$ the maps $\xi\mapsto\langle A\xi,\widehat{s}\rangle$ and $\xi\mapsto\langle\xi,B\widehat{s}\rangle$ are continuous and agree at every $\widehat{q}$, so by (P2) $\langle A\xi,\widehat{s}\rangle=\langle\xi,B\widehat{s}\rangle$ for all $\xi\in\mathcal{H}$ and $s\in\mathcal{P}_{d}$. Now for fixed $\xi$ the maps $\eta\mapsto\langle A\xi,\eta\rangle$ and $\eta\mapsto\langle\xi,B\eta\rangle$ are continuous and agree at every $\widehat{s}$, so by (P2) $\langle A\xi,\eta\rangle=\langle\xi,B\eta\rangle$ for all $\xi,\eta\in\mathcal{H}$. Thus $A$ is an adjoint of $B$, and $B^{*}=A$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-unique; that is, $L_{p}^{*}=L_{p^{*}}$ and $R_{p}^{*}=R_{p^{*}}$. If $a\in\mathcal{P}_{d,\mathrm{sa}}$, then $a^{*}=a$, so $L_{a}$ is an adjoint of $L_{a}$ and $R_{a}$ of $R_{a}$, and they are 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.

4. (Commutation) $L_{p}R_{q}$ and $R_{q}L_{p}$ belong to $\mathcal{L}(\mathcal{H})$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations, so they are continuous by (P3); and for $s\in\mathcal{P}_{d}$, by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra,

By (P2), $L_{p}R_{q}=R_{q}L_{p}$.

5. (Vacuum) By The Complex GNS Space of a Tracial State on Noncommutative Polynomials §vacuum, $\Omega=\widehat{1}$. By (P1), Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint ($1^{*}=1$), Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials and condition (a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, $\lVert\Omega\rVert^{2}=\lambda(1^{*}1)=\lambda(1)=1$, so $\lVert\Omega\rVert=1$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, $L_{p}\Omega=\widehat{p1}=\widehat{p}$ and $R_{p}\Omega=\widehat{1p}=\widehat{p}$. The identities $\langle\widehat{p},\widehat{q}\rangle=\lambda(p^{*}q)$ and $\lVert\widehat{p}\rVert=\lVert p\rVert_{\lambda}$ are (P1), and $\langle\Omega,L_{p}\Omega\rangle=\langle\widehat{1},\widehat{p}\rangle=\lambda(1^{*}p)=\lambda(p)$.

6. (Conjugation) Let $T:\mathcal{P}_{d}\to\mathcal{H}$, $Tq=\widehat{q^{*}}$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and (P1), $T$ is additive and $T(cq)=\widehat{\overline{c}\,q^{*}}=\overline{c}\,Tq$. For $u,v\in\mathcal{P}_{d}$, by (P1), Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, traciality and Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint,

In particular $\lVert Tq\rVert^{2}=\overline{h(q,q)}=h(q,q)$, as $h(q,q)$ is real (claim 1 of Properties of Complex Conjugation and Modulus). By The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §extension-antilinear with $K=\mathcal{H}$ and $C=1$, there is exactly one continuous $J_{\lambda}:\mathcal{H}\to\mathcal{H}$ with $J_{\lambda}\widehat{q}=\widehat{q^{*}}$ for every $q$; it is additive, $J_{\lambda}(c\xi)=\overline{c}\,J_{\lambda}\xi$, $\lVert J_{\lambda}\xi\rVert\le\lVert\xi\rVert$ (the norm of $\mathcal{H}$ being $|\cdot|$ by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §hilbert), and $\langle J_{\lambda}\xi,J_{\lambda}\eta\rangle=\overline{\langle\xi,\eta\rangle}=\langle\eta,\xi\rangle$ by condition 1 of Complex Inner Product Space.

The maps $J_{\lambda}J_{\lambda}$ and $J_{\lambda}L_{p}J_{\lambda}$ are additive, and complex-homogeneous since $J_{\lambda}(\overline{c}\,y)=\overline{\overline{c}}\,J_{\lambda}y=c\,J_{\lambda}y$ by claim 1 of Properties of Complex Conjugation and Modulus; and $\lVert J_{\lambda}J_{\lambda}\xi\rVert\le\lVert\xi\rVert$, $\lVert J_{\lambda}L_{p}J_{\lambda}\xi\rVert\le\lVert L_{p}\rVert_{\mathrm{op}}\lVert\xi\rVert$ 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. So both belong to $\mathcal{L}(\mathcal{H})$. For $q\in\mathcal{P}_{d}$, by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials,

By the uniqueness in claim 1, $J_{\lambda}J_{\lambda}=L_{1}$, which is $I$ by claim 2, so $J_{\lambda}J_{\lambda}\xi=\xi$; and $J_{\lambda}L_{p}J_{\lambda}=R_{p^{*}}$.