The Norm Bound of a Noncommutative Law: Multiplication by a Variable is Bounded, the Bound is Detected by Even Moments, and Laws Pull Back under Self-Adjoint Substitutions

Let $d,n\in\mathbb{N}$, with initial segments $[d]$ and $[n]$, and for $r\in\mathbb{N}$ let $\mathcal{P}_{r}=\mathbb{C}\langle x_{1},\dots,x_{r}\rangle$ be the noncommutative polynomials in $r$ variables, with product, unit $1$, variables $x_{j}$, adjoint $p\mapsto p^{*}$ and self-adjoint part $\mathcal{P}_{r,\mathrm{sa}}$. For $m\in\mathbb{N}$ write $2m=m+m$. For $a\in\mathcal{P}_{d}$ and $m\in\mathbb{N}$, $a^{m}$ is the product of $m$ factors $a$ (the product along the word of length $m$ all of whose letters are $1$, for the $1$-tuple $(a)$). Let $\lambda$ be a tracial state on $\mathcal{P}_{d}$; for $p\in\mathcal{P}_{d}$ the number $\lambda(p^{*}p)$ is real and nonnegative, and $\|p\|_{\lambda}$ denotes its nonnegative square root. Norm bounds and the sets $\Sigma_{r,R}$ are those of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound, and $\sigma_{a}$ denotes the substitution of a tuple $a$.

1. (Bounded multiplication)¶ Let $a\in\mathcal{P}_{d,\mathrm{sa}}$ and let $S>0$ be real with $\lambda(a^{2m})\le S^{2m}$ for every $m\in\mathbb{N}$ (the number $\lambda(a^{2m})$ is real by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint, as $a^{2m}$ is self-adjoint). Then $\|ap\|_{\lambda}\le S\|p\|_{\lambda}$ for every $p\in\mathcal{P}_{d}$. In particular, if $\lambda\in\Sigma_{d,R}$, then $\|x_{j}p\|_{\lambda}\le R\|p\|_{\lambda}$ for every $j\in[d]$ and $p\in\mathcal{P}_{d}$.

2. (Criterion by even moments)¶ Let $R>0$ be real. Then $\lambda\in\Sigma_{d,R}$ if and only if $\lambda(x_{j}^{2m})\le R^{2m}$ for every $j\in[d]$ and every $m\in\mathbb{N}$.

3. (Pull-back)¶ Let $a=(a_{1},\dots,a_{n})$ be an $n$-tuple in $\mathcal{P}_{d,\mathrm{sa}}$ and let $S>0$ be real with $\|a_{j}p\|_{\lambda}\le S\|p\|_{\lambda}$ for every $j\in[n]$ and $p\in\mathcal{P}_{d}$. Then $\lambda\circ\sigma_{a}:\mathcal{P}_{n}\to\mathbb{C}$ is a tracial state on $\mathcal{P}_{n}$ with norm bound $S$.

4. (Affine substitutions)¶ Let $\lambda\in\Sigma_{d,R}$ for a real $R>0$, and let $a_{j}=c_{j0}1+\sum_{k=1}^{d}c_{jk}x_{k}$ $(j\in[n])$ with real coefficients $c_{jk}$ $(0\le k\le d)$, the sum being the finite sum in the complex vector space $\mathcal{P}_{d}$. If $S>0$ is real and $|c_{j0}|+R\sum_{k=1}^{d}|c_{jk}|\le S$ for every $j\in[n]$, then $\lambda\circ\sigma_{a}\in\Sigma_{n,S}$.