Proof of The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State

Each result cited below is universally quantified over the data in its own statement. We work in the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation. Fix $n$, $R$, $H$, $\Omega$ and $T$ as in the statement. By Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §maps, sums and scalar multiples in $\mathcal{L}(H)$ are formed pointwise, so $(A+B)\Omega=A\Omega+B\Omega$ and $(aA)\Omega=a\,(A\Omega)$ for $A,B\in\mathcal{L}(H)$ and $a\in\mathbb{C}$. By Complex Hilbert Space, $H$ is a complex inner product space; its inner product is additive and homogeneous in the second argument by conditions 2 and 3 of Complex Inner Product Space.

Step 1 (linearity and normalisation). Let $p,q\in\mathcal{P}_{n}$ and $a\in\mathbb{C}$. The evaluation $p\mapsto p(T)$ is linear by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation, so $(p+q)(T)=p(T)+q(T)$ and $(ap)(T)=a\,p(T)$. Hence, by the pointwise operations and conditions 2 and 3 of Complex Inner Product Space,

so $\lambda_{T}$ is linear in the sense of Linear Map, $\mathbb{C}$ being a complex vector space over itself. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $1(T)=I$, so by Norm Induced by a Complex Inner Product and $\lVert\Omega\rVert=1$,

Step 2 (positivity). Let $p\in\mathcal{P}_{n}$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism applied to $p^{*}$ and $p$, $(p^{*}p)(T)=p^{*}(T)\,p(T)$, and since every $T_{j}$ is self-adjoint, $p^{*}(T)=p(T)^{*}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint. As $p(T)\in\mathcal{L}(H)$ and $H$ is a complex Hilbert space, $p(T)$ has an adjoint by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint, and Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus gives

By Norm Induced by a Complex Inner Product this number equals $\langle p(T)\Omega,p(T)\Omega\rangle$, which is a real number with $0\le\langle p(T)\Omega,p(T)\Omega\rangle$ by condition 4 of Complex Inner Product Space.

Step 3 (norm bound). Let $k\in\mathbb{N}$ and let $w\in W_{n}$ be a word of length $k$. By Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation, $x_{w}(T)=T_{w}$, so $\lambda_{T}(x_{w})=\langle\Omega,T_{w}\Omega\rangle$. By claim 1 of The Induced Norm is a Norm, and Induces a Metric and $\lVert\Omega\rVert=1$,

The operator $T_{w}$ lies in $\mathcal{L}(H)$ by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products; by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound, $\lVert T_{w}\rVert_{\mathrm{op}}$ is a bound for $T_{w}$, so by Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded we have $\lVert T_{w}\Omega\rVert\le\lVert T_{w}\rVert_{\mathrm{op}}\lVert\Omega\rVert=\lVert T_{w}\rVert_{\mathrm{op}}$. Since $\lVert T_{j}\rVert_{\mathrm{op}}\le R$ for every $j\in[n]$, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §bound gives $\lVert T_{w}\rVert_{\mathrm{op}}\le R^{k}$. By transitivity of $\le$ in the ordered field $\mathbb{R}$, $|\lambda_{T}(x_{w})|\le R^{k}$. Steps 1 to 3 prove Clause 1.

Step 4 (traciality on pairs of monomials). Assume from now on that $\langle\Omega,T_{u}T_{v}\Omega\rangle=\langle\Omega,T_{v}T_{u}\Omega\rangle$ for all $u,v\in W_{n}$. Let $u,v\in W_{n}$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, $x_{u}x_{v}=x_{uv}$; by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $(x_{uv})(T)=T_{uv}=T_{u}T_{v}$. Hence $\lambda_{T}(x_{u}x_{v})=\langle\Omega,T_{u}T_{v}\Omega\rangle$, and in the same way $\lambda_{T}(x_{v}x_{u})=\langle\Omega,T_{v}T_{u}\Omega\rangle$. By the assumption,

Step 5 (one monomial factor). Fix $v\in W_{n}$ and define $f,g:\mathcal{P}_{n}\to\mathbb{C}$ by $f(p)=\lambda_{T}(p\,x_{v})$ and $g(p)=\lambda_{T}(x_{v}\,p)$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, for $p,q\in\mathcal{P}_{n}$ and $a\in\mathbb{C}$ we have $(p+q)x_{v}=px_{v}+qx_{v}$, $(ap)x_{v}=a(px_{v})$, $x_{v}(p+q)=x_{v}p+x_{v}q$ and $x_{v}(ap)=a(x_{v}p)$; together with the linearity of $\lambda_{T}$ from Step 1, $f$ and $g$ are linear. By Step 4, $f(x_{u})=g(x_{u})$ for every $u\in W_{n}$. Let $c:W_{n}\to\mathbb{C}$ be the map $c(u)=f(x_{u})$. Then $f$ and $g$ are linear maps $\mathcal{P}_{n}\to\mathbb{C}$ with value $c(u)$ at $x_{u}$ for every $u$, so $f=g$ by the uniqueness in part (a) of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension. Thus $\lambda_{T}(p\,x_{v})=\lambda_{T}(x_{v}\,p)$ for all $p\in\mathcal{P}_{n}$ and $v\in W_{n}$.

Step 6 (traciality). Fix $p\in\mathcal{P}_{n}$ and define $F,G:\mathcal{P}_{n}\to\mathbb{C}$ by $F(q)=\lambda_{T}(pq)$ and $G(q)=\lambda_{T}(qp)$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, $p(q+r)=pq+pr$, $p(aq)=a(pq)$, $(q+r)p=qp+rp$ and $(aq)p=a(qp)$ for $q,r\in\mathcal{P}_{n}$ and $a\in\mathbb{C}$, so $F$ and $G$ are linear by Step 1. By Step 5, $F(x_{v})=\lambda_{T}(p\,x_{v})=\lambda_{T}(x_{v}\,p)=G(x_{v})$ for every $v\in W_{n}$, so $F=G$ by the uniqueness in part (a) of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension, applied as in Step 5 to the map $v\mapsto F(x_{v})$. Hence $\lambda_{T}(pq)=\lambda_{T}(qp)$ for all $p,q\in\mathcal{P}_{n}$.

Step 7 (conclusion). By Steps 1, 2 and 6, $\lambda_{T}$ is a linear map $\mathcal{P}_{n}\to\mathbb{C}$ satisfying conditions (a), (b) and (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state (with $d=n$), so it is a tracial state on $\mathcal{P}_{n}$. By Step 3 it has norm bound $R$ in the sense of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound. Therefore $\lambda_{T}\in\Sigma_{n,R}$, the set named in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws, which proves Clause 2.