Proof of 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

This proof uses Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint and Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §cauchy-schwarz; conditions (a), (b), (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state and Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound; The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials, The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §self-adjoint; Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint; Substitution of Noncommutative Polynomials into the Variables §substitution; Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §identity; Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words, Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation, Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid, Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §reversal; Initial Segment of the Natural Numbers and claim 1 of Basic Properties of Finite Sets; Existence and Uniqueness of Iterates of a Binary Operation and Finite Sum Notation in a Field; Linear Map; Existence and Uniqueness of the Nonnegative Square Root; claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; claims 1, 3, 4, 5 of Properties of Natural Number Powers in a Field and Addition of Exponents for Natural Number Powers in a Field; claims 1, 2, 3, 6, 8 of Properties of Complex Conjugation and Modulus and claim 3 of Canonical Form and Arithmetic of Complex Numbers; condition 1 of The Complex Numbers; claim 5 of Elementary Arithmetic in an Ordered Field and claims 2, 5, 7 of Elementary Order Arithmetic in an Ordered Field; claims 1, 2, 3(a), 3(f) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities and Real Power of a Positive Real Number; claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; Limit of a Sequence of Real Numbers, claim 3 of Arithmetic of Limits of Real Sequences and claim 1 of Order Properties of Limits of Real Sequences.

Conventions. For $m\in\mathbb{N}$, $2m$ denotes $m+m$ and $4m$ denotes $2m+2m$, computed in $\mathbb{N}$. We write $\|q\|$ for $\|q\|_{\lambda}$; thus $\|q\|\ge0$ and $\|q\|^{2}=\lambda(q^{*}q)$. Real numbers are complex numbers, and sums and products of reals formed in $\mathbb{C}$ are the real ones (condition 1 of The Complex Numbers). For real $x\ge 0$ we have $|x|=x$ by claim 8 of Properties of Complex Conjugation and Modulus. Whenever two nonnegative reals have squares in the relation $\le$ (or $=$), the reals themselves are in that relation, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (or by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root); we call this "taking square roots".

Step 1 (Seminorm facts). Let $u,v,q,r\in\mathcal{P}_{d}$ and $c\in\mathbb{C}$. (1a) $|\lambda(v^{*}u)|\le\|u\|\,\|v\|$: by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §cauchy-schwarz, $|\lambda(v^{*}u)|^{2}\le\|u\|^{2}\|v\|^{2}=(\|u\|\,\|v\|)^{2}$; take square roots. (1b) $\|q^{*}\|=\|q\|$: by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, $(q^{*})^{*}q^{*}=qq^{*}$, and $\lambda(qq^{*})=\lambda(q^{*}q)$ by (c); take square roots. (1c) $\|1\|=1$: $1^{*}1=1$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, and $\lambda(1)=1=1^{2}$ by (a). (1d) $\|cq\|=|c|\,\|q\|$: $(cq)^{*}(cq)=(\overline{c}c)\,q^{*}q=|c|^{2}q^{*}q$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and claim 3 of Properties of Complex Conjugation and Modulus; apply $\lambda$ and take square roots. (1e) $\|q+r\|\le\|q\|+\|r\|$: expanding with Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, $(q+r)^{*}(q+r)=q^{*}q+q^{*}r+r^{*}q+r^{*}r$. With $\zeta=\lambda(q^{*}r)$ we have $\lambda(r^{*}q)=\overline{\zeta}$ by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint, and $\zeta+\overline{\zeta}=2\operatorname{Re}\zeta\le2|\zeta|\le2\|q\|\,\|r\|$ by claims 2 and 6 of Properties of Complex Conjugation and Modulus and (1a). Hence $\|q+r\|^{2}\le\|q\|^{2}+2\|q\|\,\|r\|+\|r\|^{2}=(\|q\|+\|r\|)^{2}$; take square roots.

Step 2 (Powers). The initial segment $[1]$ contains $1$ (Initial Segment of the Natural Numbers) and has one element (claim 1 of Basic Properties of Finite Sets), so for each $k\in\mathbb{N}$ there is exactly one map $[k]\to[1]$, i.e. exactly one word $e_{k}\in W_{1}$ of length $k$; and $a^{k}$ is the product along $e_{k}$ for the $1$-tuple $(a)$, so $a^{k}=\sigma_{(a)}(x_{e_{k}})$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. Let $a\in\mathcal{P}_{d}$ and $k,l\in\mathbb{N}$. (2a) $a^{1}=a$: $e_{1}$ is the letter $(1)$, so $x_{e_{1}}=x_{1}$ (The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials) and $\sigma_{(a)}(x_{1})=a$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. (2b) $a^{k}a^{l}=a^{k+l}$: $e_{k}e_{l}$ has length $k+l$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid, hence equals $e_{k+l}$, and the product along $e_{k}e_{l}$ is $a^{k}a^{l}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. (2c) If $a\in\mathcal{P}_{d,\mathrm{sa}}$ then $a^{k}\in\mathcal{P}_{d,\mathrm{sa}}$: $e_{k}^{\mathrm{rev}}$ has length $k$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §reversal, hence equals $e_{k}$, so $x_{e_{k}}^{*}=x_{e_{k}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint; and $\sigma_{(a)}$ maps $\mathcal{P}_{1,\mathrm{sa}}$ into $\mathcal{P}_{d,\mathrm{sa}}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint. In particular $\lambda(a^{k})$ 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, so the hypothesis of 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 §multiplication is meaningful. (2d) If $a\in\mathcal{P}_{d,\mathrm{sa}}$ and $q,r\in\mathcal{P}_{d}$, then $(a^{k}q)^{*}(a^{l}r)=q^{*}a^{k+l}r$, by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, (2c), Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and (2b). In particular $\|a^{k}q\|^{2}=\lambda(q^{*}a^{2k}q)$.

Step 3 (Bounded multiplication). Let $a$ and $S$ be as in 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 §multiplication, and fix $p\in\mathcal{P}_{d}$. Put $c_{*}=\lambda(p^{*}p)=\|p\|^{2}$, $C=\|pp^{*}\|$, and for $n\in\mathbb{N}$, $c_{n}=\lambda(p^{*}a^{2n}p)=\|a^{n}p\|^{2}$ (by (2d)); all are real and $\ge0$.

(3a) $c_{1}^{2}\le c_{*}c_{2}$ and $c_{n+1}^{2}\le c_{n}c_{n+2}$ for $n\in\mathbb{N}$. Indeed, (1a) with $u=a^{2}p$, $v=p$ gives $c_{1}=|\lambda(p^{*}a^{2}p)|\le\|a^{2}p\|\,\|p\|$, and $\|a^{2}p\|^{2}=c_{2}$ by (2d); squaring (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field) gives the first inequality. For the second, (1a) with $u=a^{n+2}p$, $v=a^{n}p$ and (2d) give $|\lambda(p^{*}a^{n+(n+2)}p)|\le\|a^{n+2}p\|\,\|a^{n}p\|$; since $n+(n+2)=2(n+1)$ in $\mathbb{N}$, the left side is $c_{n+1}$, and the right side is $\sqrt{c_{n+2}}\sqrt{c_{n}}$; square.

(3b) $c_{n}\le C\,(S^{2})^{n}$ for $n\in\mathbb{N}$. By (c) and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, $c_{n}=\lambda\bigl(p^{*}(a^{2n}p)\bigr)=\lambda\bigl((a^{2n}p)p^{*}\bigr)=\lambda\bigl((a^{2n})^{*}(pp^{*})\bigr)$, using $(a^{2n})^{*}=a^{2n}$ from (2c). By (1a) with $u=pp^{*}$, $v=a^{2n}$, $c_{n}\le C\|a^{2n}\|$. By (2d) with $q=1$, (2b) and the hypothesis with $m=2n$, $\|a^{2n}\|^{2}=\lambda(a^{4n})\le S^{4n}=(S^{2n})^{2}$, the last by Addition of Exponents for Natural Number Powers in a Field; since $S^{2n}\ge0$ (claim 5 of Properties of Natural Number Powers in a Field), $\|a^{2n}\|\le S^{2n}$. Finally $S^{2n}=S^{n}S^{n}=(SS)^{n}=(S^{2})^{n}$ by Addition of Exponents for Natural Number Powers in a Field, claim 3 of Properties of Natural Number Powers in a Field, and $S^{2}=S^{1}S^{1}=SS$ (the same lemma and claim 1 there). Multiplying by $C\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives (3b).

(3c) $c_{1}\le S^{2}c_{*}$. If $c_{*}=0$, then $0\le c_{1}^{2}\le0$ by (3a), so $c_{1}^{2}=0^{2}$ and $c_{1}=0$ by taking square roots. If $c_{1}=0$ the claim holds since $S^{2}c_{*}\ge0$. So let $c_{*}>0$ and $c_{1}>0$, and put $\rho=c_{1}/c_{*}$, which is $>0$ by claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field. We show by induction on $n\in\mathbb{N}$ the statement $T(n)$: $c_{n}>0$ and $c_{n+1}\ge\rho c_{n}$. For $T(1)$: $c_{1}>0$, and $c_{*}(\rho c_{1})=c_{1}^{2}\le c_{*}c_{2}$ by (3a); multiplying by $c_{*}^{-1}>0$ gives $\rho c_{1}\le c_{2}$. If $T(n)$ holds, then $c_{n+1}\ge\rho c_{n}>0$ (claims 5 and 2 of Elementary Order Arithmetic in an Ordered Field); multiplying $\rho c_{n}\le c_{n+1}$ by $c_{n+1}\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field) and using (3a), $c_{n}(\rho c_{n+1})\le c_{n+1}^{2}\le c_{n}c_{n+2}$, and multiplying by $c_{n}^{-1}>0$ gives $\rho c_{n+1}\le c_{n+2}$; so $T(n+1)$ holds. Next, $c_{n}\ge\rho^{n}c_{*}$ for all $n\in\mathbb{N}$, by induction: $c_{1}=\rho c_{*}=\rho^{1}c_{*}$, and $c_{n+1}\ge\rho c_{n}\ge\rho\,\rho^{n}c_{*}=\rho^{n+1}c_{*}$ by $T(n)$, $\rho\ge0$ and claim 1 of Properties of Natural Number Powers in a Field. Combining with (3b), $\rho^{n}c_{*}\le C(S^{2})^{n}$ for all $n$; for $n=1$ this gives $0<c_{1}\le CS^{2}$, hence $C>0$. Put $t=\rho/S^{2}>0$ and $D=C/c_{*}>0$. Then $\rho^{n}=t^{n}(S^{2})^{n}$ by claim 3 of Properties of Natural Number Powers in a Field, and $(S^{2})^{n}>0$ by claims 4 and 5 there, so multiplying by $\bigl((S^{2})^{n}c_{*}\bigr)^{-1}>0$ gives

Regard $n\in\mathbb{N}$ as the real number $\iota(n)>0$ (claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field), as in the setting of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities; by claim 1 there the real power $t^{n}$ is the natural power. If $D^{1/n}<t$, claim 2 there (with exponent $n$ and $x=D$) gives $t^{n}>D$, which is impossible; hence $t\le D^{1/n}$ for every $n$, the order being total. By Real Power of a Positive Real Number, $D^{1/n}=\exp\bigl(n^{-1}\log D\bigr)$, where $n^{-1}=1/n$ by claim 1 there. By claim 3(a) there (with $a=1$), $n^{-1}\to0$; by claim 3 of Arithmetic of Limits of Real Sequences, $n^{-1}\log D\to0$; by claim 3(f) there, $D^{1/n}\to\exp(0)$, and $\exp(0)=\exp(0\cdot\log D)=D^{0}=1$ by claim 1 there. The constant sequence $t$ converges to $t$ directly by Limit of a Sequence of Real Numbers, so claim 1 of Order Properties of Limits of Real Sequences gives $t\le1$, i.e. $\rho\le S^{2}$, and $c_{1}=\rho c_{*}\le S^{2}c_{*}$.

(3d) By (2d) and (2a), $\|ap\|^{2}=c_{1}\le S^{2}c_{*}=(S\|p\|)^{2}$; since $S\|p\|\ge0$, taking square roots gives $\|ap\|\le S\|p\|$. This proves the first assertion of 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 §multiplication.

Step 4 (Variables). Let $j\in[d]$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint, $x_{j}\in\mathcal{P}_{d,\mathrm{sa}}$. For $k\in\mathbb{N}$ let $f_{k}$ be the word of length $k$ all of whose letters are $j$. Then $x_{j}^{k}=x_{f_{k}}$, by induction on $k$: $x_{j}^{1}=x_{j}=x_{f_{1}}$ by (2a) and The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials; and $x_{j}^{k+1}=x_{j}^{k}x_{j}^{1}=x_{f_{k}}x_{(j)}=x_{f_{k}(j)}=x_{f_{k+1}}$ by (2b), Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials and Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation. Now let $\lambda\in\Sigma_{d,R}$ and $m\in\mathbb{N}$. The number $\lambda(x_{j}^{2m})$ is real by (2c), so it equals its real part (claim 3 of Canonical Form and Arithmetic of Complex Numbers), and by claim 6 of Properties of Complex Conjugation and Modulus and Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound (as $f_{2m}$ has length $2m$),

Thus Step 3 applies with $a=x_{j}$ and $S=R$, giving $\|x_{j}p\|\le R\|p\|$ for every $p$. This proves the second assertion of 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 §multiplication, and the displayed bound is the "only if" part of 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 §criterion.

Step 5 (Products along words). Let $n\in\mathbb{N}$, let $b=(b_{1},\dots,b_{n})$ be an $n$-tuple in $\mathcal{P}_{d,\mathrm{sa}}$, and let $T>0$ be real with $\|b_{j}q\|\le T\|q\|$ for all $j\in[n]$, $q\in\mathcal{P}_{d}$. We show by induction on $k\in\mathbb{N}$ that $\|b_{w}\|\le T^{k}$ for every $w\in W_{n}$ of length $k$. If $k=1$, $w$ is a letter $(j)$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, $b_{w}=\sigma_{b}(x_{j})=b_{j}=b_{j}1$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, so $\|b_{w}\|\le T\|1\|=T=T^{1}$ by (1c). If $w$ has length $k+1$ (the successor of $k$), then $w=w'(j)$ with $w'$ of length $k$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, and $b_{w}=b_{w'}b_{j}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. By (1b), Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and $b_{j}^{*}=b_{j}$,

using the induction hypothesis, claim 5 of Elementary Arithmetic in an Ordered Field and claim 1 of Properties of Natural Number Powers in a Field. Consequently, by (1a), (1c) and $1^{*}b_{w}=b_{w}$,

Step 6 (Criterion, "if"). Suppose $\lambda(x_{j}^{2m})\le R^{2m}$ for all $j\in[d]$, $m\in\mathbb{N}$. By Step 3 with $a=x_{j}$ and $S=R$, $\|x_{j}q\|\le R\|q\|$ for all $j$ and $q$. Apply Step 5 with $n=d$, $T=R$ and $b$ the tuple $x=(x_{1},\dots,x_{d})$ of variables: its product along $w$ is $\sigma_{x}(x_{w})=x_{w}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §identity. Hence $|\lambda(x_{w})|\le R^{k}$ for every word $w$ of length $k$, i.e. $\lambda\in\Sigma_{d,R}$. With Step 4 this proves 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 §criterion.

Step 7 (Pull-back). Let $a$ and $S$ be as in 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 §pullback and $\mu=\lambda\circ\sigma_{a}$. It is linear, directly from Linear Map, as $\sigma_{a}$ (Substitution of Noncommutative Polynomials into the Variables §substitution) and $\lambda$ are. For $p,q\in\mathcal{P}_{n}$: $\mu(1)=\lambda(1)=1$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and (a); $\mu(p^{*}p)=\lambda\bigl(\sigma_{a}(p)^{*}\sigma_{a}(p)\bigr)$ is real and $\ge0$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint (the $a_{j}$ being self-adjoint) and (b); and $\mu(pq)=\lambda\bigl(\sigma_{a}(p)\sigma_{a}(q)\bigr)=\lambda\bigl(\sigma_{a}(q)\sigma_{a}(p)\bigr)=\mu(qp)$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism and (c). So $\mu$ is a tracial state on $\mathcal{P}_{n}$. For $w\in W_{n}$ of length $k$, $\mu(x_{w})=\lambda(a_{w})$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, and $|\lambda(a_{w})|\le S^{k}$ by Step 5 with $b=a$, $T=S$. Hence $\mu$ has norm bound $S$ (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound with $n$ in place of $d$). This proves 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 §pullback.

Step 8 (Affine substitutions). Let $\lambda$, $R$, $a_{j}$, $c_{jk}$ and $S$ be as in 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 §affine, and fix $j\in[n]$. The sum $\sum_{k=1}^{d}c_{jk}x_{k}$ in $\mathcal{P}_{d}$ is, by Finite Sum Notation in a Vector Space, the iterate of Existence and Uniqueness of Iterates of a Binary Operation for the addition of $\mathcal{P}_{d}$: its partial sums satisfy $s_{1}=c_{j1}x_{1}$ and $s_{l+1}=s_{l}+c_{j,l+1}x_{l+1}$, and it equals $s_{d}$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint ($1$ and the $x_{k}$ are self-adjoint, and $\mathcal{P}_{d,\mathrm{sa}}$ is closed under sums and real multiples), each $s_{l}$ is self-adjoint by induction on $l$, and so is $a_{j}=c_{j0}1+s_{d}$. Let $q\in\mathcal{P}_{d}$ and put $\gamma_{l}=\sum_{k=1}^{l}|c_{jk}|$ (Finite Sum Notation in a Field). By induction on $l\in[d]$, $\|s_{l}q\|\le R\gamma_{l}\|q\|$: for $l=1$, $\|c_{j1}x_{1}q\|=|c_{j1}|\,\|x_{1}q\|\le R|c_{j1}|\,\|q\|$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, (1d) and Step 4; and $s_{l+1}q=s_{l}q+c_{j,l+1}(x_{l+1}q)$, so by (1e), (1d), Step 4 and $\gamma_{l+1}=\gamma_{l}+|c_{j,l+1}|$,

Hence, as $a_{j}q=c_{j0}q+s_{d}q$, (1e) and (1d) give $\|a_{j}q\|\le(|c_{j0}|+R\gamma_{d})\|q\|\le S\|q\|$ (claim 5 of Elementary Arithmetic in an Ordered Field, $\|q\|\ge0$). Now Step 7 applies to the $n$-tuple $a$ of self-adjoint polynomials and gives that $\lambda\circ\sigma_{a}$ is a tracial state on $\mathcal{P}_{n}$ with norm bound $S$, i.e. $\lambda\circ\sigma_{a}\in\Sigma_{n,S}$. This proves 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 §affine.