Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound

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 $1$, monomials $x_{w}$ and adjoint $p\mapsto p^{*}$; $\mathcal{P}_{d}$ 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 $\mathbb{C}$ be the complex numbers, containing the real numbers $\mathbb{R}$, with modulus $|\cdot|$, and for real $R>0$ and $k\in\mathbb{N}$ let $R^{k}$ be the natural power.

1. (Tracial states)¶ A tracial state on $\mathcal{P}_{d}$ is a linear map $\lambda:\mathcal{P}_{d}\to\mathbb{C}$ such that, for all $p,q\in\mathcal{P}_{d}$,

(a) $\lambda(1)=1$;

(b) $\lambda(p^{*}p)$ is a real number and $\lambda(p^{*}p)\ge0$;

(c) $\lambda(pq)=\lambda(qp)$.

2. (Norm bound)¶ Let $R>0$ be real. A tracial state $\lambda$ on $\mathcal{P}_{d}$ has norm bound $R$ if $|\lambda(x_{w})|\le R^{k}$ for every $k\in\mathbb{N}$ and every word $w$ of length $k$ in the letters $1,\dots,d$. The set of tracial states on $\mathcal{P}_{d}$ with norm bound $R$ is written $\Sigma_{d,R}$.

3. (Laws)¶ A noncommutative law of $d$ variables is a tracial state on $\mathcal{P}_{d}$ that has norm bound $R$ for some real $R>0$. The set of noncommutative laws of $d$ variables is written $\Sigma_{d}$; thus $\Sigma_{d}=\bigcup_{R>0}\Sigma_{d,R}$.