Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples

Let $d\in\mathbb{N}$, let $\mathcal{P}_{d}=\mathbb{C}\langle x_{1},\dots,x_{d}\rangle$ be the noncommutative polynomials in $d$ variables, with unit $1$, empty word $\varnothing$, adjoint $p\mapsto p^{*}$ and self-adjoint part $\mathcal{P}_{d,\mathrm{sa}}$, a real vector space by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. Let $\mathbb{C}$ be the complex numbers, containing $\mathbb{R}$, with imaginary unit $i$, conjugation $z\mapsto\overline{z}$ and modulus $|\cdot|$. Let $\lambda$ be a tracial state on $\mathcal{P}_{d}$.

1. (Adjoints)¶ For all $p,q\in\mathcal{P}_{d}$: $\lambda(q^{*}p)=\overline{\lambda(p^{*}q)}$ and $\lambda(p^{*})=\overline{\lambda(p)}$. In particular $\lambda(a)$ is real for every $a\in\mathcal{P}_{d,\mathrm{sa}}$.

2. (Self-adjoint pairing)¶ For all $a,b\in\mathcal{P}_{d,\mathrm{sa}}$ the number $\lambda(ab)$ is real, and the map $\beta_{\lambda}:\mathcal{P}_{d,\mathrm{sa}}\times\mathcal{P}_{d,\mathrm{sa}}\to\mathbb{R}$, $\beta_{\lambda}(a,b)=\lambda(ab)$, is symmetric, bilinear over $\mathbb{R}$ and positive semidefinite, in the sense of Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences.

3. (Cauchy-Schwarz)¶ For all $p,q\in\mathcal{P}_{d}$: $|\lambda(q^{*}p)|^{2}\le\lambda(p^{*}p)\,\lambda(q^{*}q)$.

4. (Real and imaginary parts)¶ If $p=a+ib$ with $a,b\in\mathcal{P}_{d,\mathrm{sa}}$, then $\lambda(p^{*}p)=\lambda(a^{2})+\lambda(b^{2})$.

5. (Monotonicity in the bound)¶ If $0<R\le R'$ are real, then $\Sigma_{d,R}\subseteq\Sigma_{d,R'}$, where $\Sigma_{d,R}$ is the set of tracial states with norm bound $R$.

6. (The law of the zero tuple)¶ The map $\delta:\mathcal{P}_{d}\to\mathbb{C}$, $\delta(p)=p(\varnothing)$, belongs to $\Sigma_{d,R}$ for every real $R>0$. In particular $\Sigma_{d,R}$ is nonempty.