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

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $d\in\mathbb{N}$, let $r>0$ be real and let $\lambda\in\Sigma_{d,r}$. Let $\mathcal{H}=\mathcal{H}_{\lambda}$ be the complex GNS space of $\lambda$, with inner product $\langle\cdot,\cdot\rangle$ and norm $\lVert\cdot\rVert$, let $\widehat{p}$ be the class of $p\in\mathcal{P}_{d}$, and let $\Omega=\Omega_{\lambda}$ be the vacuum vector.

1. (Multiplication operators)¶ For every $p\in\mathcal{P}_{d}$ there are exactly one $L_{p}\in\mathcal{L}(\mathcal{H})$ and exactly one $R_{p}\in\mathcal{L}(\mathcal{H})$ with

Moreover $\lVert L_{x_{j}}\rVert_{\mathrm{op}}\le r$ and $\lVert R_{x_{j}}\rVert_{\mathrm{op}}\le r$ for every $j\in[d]$.

2. (Algebra rules)¶ For all $p,q\in\mathcal{P}_{d}$ and $c\in\mathbb{C}$:

3. (Adjoints)¶ For every $p\in\mathcal{P}_{d}$: $L_{p}^{*}=L_{p^{*}}$ and $R_{p}^{*}=R_{p^{*}}$. In particular $L_{a}$ and $R_{a}$ are self-adjoint for $a\in\mathcal{P}_{d,\mathrm{sa}}$.

4. (Commutation)¶ $L_{p}R_{q}=R_{q}L_{p}$ for all $p,q\in\mathcal{P}_{d}$.

5. (Vacuum)¶ $\lVert\Omega\rVert=1$, and for all $p,q\in\mathcal{P}_{d}$: $L_{p}\Omega=R_{p}\Omega=\widehat{p}$, $\langle\widehat{p},\widehat{q}\rangle=\lambda(p^{*}q)$, $\lVert\widehat{p}\rVert=\lVert p\rVert_{\lambda}$ and $\langle\Omega,L_{p}\Omega\rangle=\lambda(p)$.

6. (Conjugation)¶ There is exactly one continuous map $J_{\lambda}:\mathcal{H}\to\mathcal{H}$ with $J_{\lambda}\widehat{p}=\widehat{p^{*}}$ for every $p\in\mathcal{P}_{d}$. It satisfies, for all $\xi,\eta\in\mathcal{H}$, $c\in\mathbb{C}$ and $p\in\mathcal{P}_{d}$,