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

This proof uses 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 §linear, The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials, The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §product, The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §adjoint; Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §vector-space, 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, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §decomposition; the notion of a symmetric, bilinear, positive semidefinite form from Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences; Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words, Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §reversal, Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §factorisations; Sum over a Finite Index Set, Finite Sum Notation in a Field, claim 2 of Basic Properties of Finite Sets; conditions 1 and 2 of The Complex Numbers, Complex Conjugate, Modulus of a Complex Number; claims 1 and 3 of Canonical Form and Arithmetic of Complex Numbers; claims 1 and 3 of Properties of Complex Conjugation and Modulus; claim 5 of Elementary Arithmetic in an Ordered Field and claims 6, 8 of Elementary Order Arithmetic in an Ordered Field; claim 5 of Properties of Natural Number Powers in a Field.

Conventions. By condition 1 of The Complex Numbers and claim 1 of Canonical Form and Arithmetic of Complex Numbers, sums, differences, products and quotients of real numbers formed in $\mathbb{C}$ are the real ones; in particular they are real. Products of polynomials are expanded with the associative, distributive and scalar rules of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and the vector-space laws of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §vector-space, and $\lambda$ is then applied termwise, being linear. Since $i=0+1\cdot i$, claim 3 of Canonical Form and Arithmetic of Complex Numbers gives $\operatorname{Re}i=0$ and $\operatorname{Im}i=1$, so $\overline{i}=-i$ by Complex Conjugate; and $i\cdot i=-1$ by condition 2 of The Complex Numbers, so $(-i)\,i=1$. A real number $x$ satisfies $\overline{x}=x$ by claim 1 of Properties of Complex Conjugation and Modulus.

Step 1 (Adjoints). Let $p,q\in\mathcal{P}_{d}$ and put $\alpha=\lambda(p^{*}p)$, $\beta=\lambda(q^{*}q)$, which are real and nonnegative by (b), and $z=\lambda(p^{*}q)$, $w=\lambda(q^{*}p)$. By the rules $(p+q)^{*}=p^{*}+q^{*}$ and $(cp)^{*}=\overline{c}\,p^{*}$ of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and $\overline{i}=-i$,

using $(-i)i=1$ in the last term. Applying $\lambda$, the numbers $\alpha+\beta+(z+w)$ and $\alpha+\beta+i(z-w)$ are real by (b); subtracting the real number $\alpha+\beta$, the numbers $s=z+w$ and $t=i(z-w)$ are real. By claim 1 of Properties of Complex Conjugation and Modulus (conjugation is additive and multiplicative, and fixes reals), $\overline{z}+\overline{w}=\overline{s}=s=z+w$ and $-i(\overline{z}-\overline{w})=\overline{t}=t=i(z-w)$. Multiplying the second identity by $i$ and using $i(-i)=1$, $i\cdot i=-1$ gives $\overline{z}-\overline{w}=w-z$. Adding this to the first identity gives $2\overline{z}=2w$, where $2=1+1$ is a nonzero real number by claim 8 of Elementary Order Arithmetic in an Ordered Field; hence $w=\overline{z}$, that is, $\lambda(q^{*}p)=\overline{\lambda(p^{*}q)}$.

Now let $p\in\mathcal{P}_{d}$ and apply this identity to the pair $(1,p)$ in place of $(p,q)$: $\lambda(p^{*}1)=\overline{\lambda(1^{*}p)}$. Since $1^{*}=1$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and $p^{*}1=p^{*}$, $1p=p$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, this reads $\lambda(p^{*})=\overline{\lambda(p)}$. If $a\in\mathcal{P}_{d,\mathrm{sa}}$, then $a^{*}=a$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §self-adjoint, so $\lambda(a)=\lambda(a^{*})=\overline{\lambda(a)}$, and $\lambda(a)$ is real by claim 1 of Properties of Complex Conjugation and Modulus. This proves Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint.

Step 2 (Self-adjoint pairing). Let $a,b\in\mathcal{P}_{d,\mathrm{sa}}$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, $(ab)^{*}=b^{*}a^{*}=ba$, so by Step 1 and condition (c), $\overline{\lambda(ab)}=\lambda((ab)^{*})=\lambda(ba)=\lambda(ab)$; thus $\lambda(ab)$ is real by claim 1 of Properties of Complex Conjugation and Modulus, and $\beta_{\lambda}$ is a well-defined map into $\mathbb{R}$ on the real vector space $\mathcal{P}_{d,\mathrm{sa}}$ of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. It is symmetric: $\beta_{\lambda}(a,b)=\lambda(ab)=\lambda(ba)=\beta_{\lambda}(b,a)$ by (c). For fixed $a$, the map $b\mapsto\beta_{\lambda}(a,b)$ is linear over $\mathbb{R}$: for $b,b'\in\mathcal{P}_{d,\mathrm{sa}}$ and real $c$, $a(b+b')=ab+ab'$ and $a(cb)=c(ab)$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, and $\lambda$ is linear. By symmetry it is also linear in the first variable, so $\beta_{\lambda}$ is bilinear. Finally $\beta_{\lambda}(a,a)=\lambda(aa)=\lambda(a^{*}a)\ge0$ by (b). These are exactly the three conditions (symmetric, bilinear, positive semidefinite) imposed on $\beta$ in Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences, which proves Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §pairing.

Step 3 (Cauchy-Schwarz). Let $p,q\in\mathcal{P}_{d}$, and put $\alpha=\lambda(p^{*}p)\ge0$, $\beta=\lambda(q^{*}q)\ge0$ (real by (b)) and $z=\lambda(q^{*}p)$; by Step 1, $\lambda(p^{*}q)=\overline{z}$. If $z=0$, then $|z|=0$ by claim 3 of Properties of Complex Conjugation and Modulus, and $0=\beta\cdot0\le\beta\alpha$ by claim 5 of Elementary Arithmetic in an Ordered Field, which is the assertion. Suppose $z\neq0$. Then $|z|\ge0$ by Modulus of a Complex Number and $|z|\ne0$ by claim 3 of Properties of Complex Conjugation and Modulus, so $|z|>0$. Put $u=|z|^{-1}z$; by claims 1 and 3 of Properties of Complex Conjugation and Modulus, $\overline{u}=|z|^{-1}\overline{z}$, and

For real $t$ let $r=p-t\,u\,q$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, $r^{*}=p^{*}-t\,\overline{u}\,q^{*}$, and expanding,

which is real and nonnegative by (b). If $\beta=0$, the choice $t=(\alpha+1)/(2|z|)$ gives $0\le\alpha-(\alpha+1)=-1$, contradicting $0<1$ (claim 6 of Elementary Order Arithmetic in an Ordered Field; by claims 4 and 2 of the same lemma, $0<1$ gives $-1<0$, which is incompatible with $0\le-1$). Hence $\beta>0$, and the choice $t=|z|/\beta$ gives $0\le\alpha-|z|^{2}/\beta$; multiplying by $\beta\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $0\le\alpha\beta-|z|^{2}$, that is, $|\lambda(q^{*}p)|^{2}\le\lambda(p^{*}p)\,\lambda(q^{*}q)$. This proves Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §cauchy-schwarz.

Step 4 (Real and imaginary parts). Let $p=a+ib$ with $a,b\in\mathcal{P}_{d,\mathrm{sa}}$. By the uniqueness in Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §decomposition, this is the decomposition of that clause, so $p^{*}p=a^{2}+b^{2}+i(ab-ba)$, where $a^{2}=aa$ and $b^{2}=bb$. Applying the linear map $\lambda$ and using $\lambda(ab)=\lambda(ba)$ from (c), $\lambda(p^{*}p)=\lambda(a^{2})+\lambda(b^{2})+i(\lambda(ab)-\lambda(ba))=\lambda(a^{2})+\lambda(b^{2})$. This proves Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §decomposition.

Step 5 (Monotonicity). Let $0<R\le R'$ and $\lambda\in\Sigma_{d,R}$. For $k\in\mathbb{N}$ and a word $w$ of length $k$, Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound gives $|\lambda(x_{w})|\le R^{k}$, and $R^{k}\le R'^{k}$ by claim 5 of Properties of Natural Number Powers in a Field (as $0\le R\le R'$). Hence $|\lambda(x_{w})|\le R'^{k}$, and $\lambda\in\Sigma_{d,R'}$. This proves Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone.

Step 6 (The law of the zero tuple). Let $R>0$ be real. The map $\delta$ is linear, since $(p+q)(\varnothing)=p(\varnothing)+q(\varnothing)$ and $(cp)(\varnothing)=c\,p(\varnothing)$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §linear. By The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials, $\delta(1)=x_{\varnothing}(\varnothing)=1$, which is (a). For $p,q\in\mathcal{P}_{d}$, The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §product gives $(pq)(\varnothing)=\sum_{(u,v)\in F(\varnothing)}p(u)q(v)$, and $F(\varnothing)=\{(\varnothing,\varnothing)\}$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §factorisations. This set has $1$ element by claim 2 of Basic Properties of Finite Sets; with the bijection $\varphi:[1]\to F(\varnothing)$, $\varphi(1)=(\varnothing,\varnothing)$, Sum over a Finite Index Set and Finite Sum Notation in a Field give that the sum is its single term:

Since $\mathbb{C}$ is commutative, $\delta(pq)=q(\varnothing)p(\varnothing)=\delta(qp)$, which is (c). By The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §adjoint and $\varnothing^{\mathrm{rev}}=\varnothing$ (Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §reversal), $\delta(p^{*})=\overline{p(\varnothing)}$, hence $\delta(p^{*}p)=\overline{p(\varnothing)}\,p(\varnothing)=|p(\varnothing)|^{2}$ by claim 3 of Properties of Complex Conjugation and Modulus; this is a real number, and it is $\ge0$ as the product of the nonnegative real number $|p(\varnothing)|$ (Modulus of a Complex Number) with itself (claim 5 of Elementary Arithmetic in an Ordered Field). This is (b), so $\delta$ is a tracial state. Finally let $k\in\mathbb{N}$ and $w$ a word of length $k$. By Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words the empty word has no length, so $w\ne\varnothing$ and $\delta(x_{w})=x_{w}(\varnothing)=0$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials. Hence $|\delta(x_{w})|=0\le R^{k}$, using claim 3 of Properties of Complex Conjugation and Modulus and claim 5 of Properties of Natural Number Powers in a Field ($0\le R$). Thus $\delta\in\Sigma_{d,R}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound, and in particular $\Sigma_{d,R}\ne\emptyset$. This proves Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §zero-law.