The GNS Space of a Noncommutative Law

Let $d\in\mathbb{N}$, let $\mathcal{P}_{d,\mathrm{sa}}$ be the real vector space of self-adjoint noncommutative polynomials in $d$ variables (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint), and let $\lambda\in\Sigma_{d}$ be a noncommutative law of $d$ variables. Let $\beta_{\lambda}(a,b)=\lambda(ab)$ be its self-adjoint pairing, a symmetric, bilinear, positive semidefinite map on $\mathcal{P}_{d,\mathrm{sa}}$.

1. (GNS space)¶ The GNS space of $\lambda$ is the Hilbert completion $L^{2}(\lambda)$ of $(\mathcal{P}_{d,\mathrm{sa}},\beta_{\lambda})$, with inner product $\langle\cdot,\cdot\rangle_{\lambda}$ and norm $|\cdot|_{\lambda}$.

2. (Classes of polynomials)¶ For $a\in\mathcal{P}_{d,\mathrm{sa}}$, the class of $a$ is $[a]_{\lambda}=J(a)\in L^{2}(\lambda)$, where $J$ is the canonical map of the completion; thus $\langle[a]_{\lambda},[b]_{\lambda}\rangle_{\lambda}=\lambda(ab)$ for all $a,b\in\mathcal{P}_{d,\mathrm{sa}}$ by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §isometry.