Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the natural numbers, $d\in\mathbb{N}$, $W_{2d}$ and the adjoint words $w\mapsto w^{*}$ as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words, and $\mathcal{L}_{d}$ the set of unitary laws of $d$-tuples. Word polynomials, the word polynomials $e_{w}$, their products, adjoints, evaluations $\lambda(X)$ and $\ell^{1}$ norms $\lVert X\rVert_{1}$ are those of Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions. Let $\mathbb{C}$ be the complex numbers with conjugation $z\mapsto\overline{z}$ and modulus $|z|$. Let $\lambda\in\mathcal{L}_{d}$, let $X,Y,Z$ be word polynomials and $z\in\mathbb{C}$.

Then the following hold.

1. (Algebra)¶ $(X+Y)Z=XZ+YZ$, $Z(X+Y)=ZX+ZY$, $(zX)Y=z(XY)=X(zY)$, $(X+Y)^{*}=X^{*}+Y^{*}$, $(zX)^{*}=\overline{z}\,X^{*}$, $(XY)^{*}=Y^{*}X^{*}$, $(X^{*})^{*}=X$, $(XY)Z=X(YZ)$, $e_{\varnothing}X=Xe_{\varnothing}=X$, and $e_{u}e_{v}=e_{uv}$, $(e_{u})^{*}=e_{u^{*}}$ for all $u,v\in W_{2d}$.

2. (Linearity)¶ $\lambda(X+Y)=\lambda(X)+\lambda(Y)$ and $\lambda(zX)=z\,\lambda(X)$.

3. (Traciality)¶ $\lambda(XY)=\lambda(YX)$.

4. (Adjoints)¶ $\lambda(X^{*})=\overline{\lambda(X)}$.

5. (Positivity)¶ $\lambda(X^{*}X)$ is a nonnegative real number. If $X$ and $Y$ are self-adjoint, then $\lambda(XY)$ is a real number and $\lambda(XX)\ge0$.

6. (Cauchy–Schwarz)¶ $|\lambda(X^{*}Y)|^{2}\le\lambda(X^{*}X)\,\lambda(Y^{*}Y)$.

7. ($\ell^{1}$ bounds)¶ $|\lambda(X)|\le\lVert X\rVert_{1}$, $\lVert X+Y\rVert_{1}\le\lVert X\rVert_{1}+\lVert Y\rVert_{1}$, $\lVert zX\rVert_{1}=|z|\,\lVert X\rVert_{1}$, $\lVert X^{*}\rVert_{1}=\lVert X\rVert_{1}$ and $\lVert XY\rVert_{1}\le\lVert X\rVert_{1}\lVert Y\rVert_{1}$.