The Complex GNS Space of a Tracial State on Noncommutative Polynomials

Let $d\in\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers, and let $\mathcal{P}_{d}=\mathbb{C}\langle x_{1},\dots,x_{d}\rangle$ be the noncommutative polynomials in $d$ variables, with product, unit $1$ and adjoint $p\mapsto p^{*}$; it is a complex vector space by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §vector-space. Let $\lambda$ be a tracial state on $\mathcal{P}_{d}$ and let $h_{\lambda}(p,q)=\lambda(p^{*}q)$ for $p,q\in\mathcal{P}_{d}$. The map $h_{\lambda}$ satisfies the hypotheses of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §real-structure: it is additive and complex-homogeneous in $q$ because the product is bilinear by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and $\lambda$ is linear; $h_{\lambda}(q,p)=\overline{h_{\lambda}(p,q)}$ by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint; and $h_{\lambda}(p,p)$ is real and nonnegative by condition (b) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state.

1. (GNS space)¶ The complex GNS space of $\lambda$ is the complex Hilbert completion $\mathcal{H}_{\lambda}$ of $(\mathcal{P}_{d},h_{\lambda})$, which is 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. Its inner product $\langle\cdot,\cdot\rangle_{\mathcal{H}_{\lambda}}$ is the pairing $\langle\cdot,\cdot\rangle_{h_{\lambda}}$ of that definition, and $\lVert\cdot\rVert_{\mathcal{H}_{\lambda}}$ is the induced norm.

2. (Classes)¶ For $p\in\mathcal{P}_{d}$ the class of $p$ is $\widehat{p}=J_{h_{\lambda}}(p)\in\mathcal{H}_{\lambda}$, where $J_{h_{\lambda}}$ is the canonical map; when the state must be named it is written $\widehat{p}^{\,\lambda}$.

3. (Vacuum)¶ The vacuum vector of $\lambda$ is $\Omega_{\lambda}=\widehat{1}$.