Proof of The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications

Each result cited below is universally quantified over the data in its own statement.

Write $\mathcal{H}=\mathcal{H}_{\lambda}$, $\Omega=\Omega_{\lambda}$, $J=J_{\lambda}$, $\mathcal{A}=\mathcal{A}_{\lambda}$, and $\widehat{p}$ for the class of $p\in\mathcal{P}_{d}$. By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law there is a real $r>0$ with $\lambda\in\Sigma_{d,r}$, so Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation applies to $\lambda$; by Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §gns, $L_{p}$, $R_{p}$ and $J$ are the operators of that lemma.

Claim 1. We check the conditions of Cyclic Tracial Operator Algebras and Their Traces §triple. Condition (a): $\mathcal{A}\subseteq\mathcal{L}(\mathcal{H})$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §multiplication; $I=L_{1}\in\mathcal{A}$, and for $p,q\in\mathcal{P}_{d}$ and $c\in\mathbb{C}$, $L_{p}+L_{q}=L_{p+q}$, $cL_{p}=L_{cp}$, $L_{p}L_{q}=L_{pq}$ and $L_{p}^{*}=L_{p^{*}}$ belong to $\mathcal{A}$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §algebra and Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §adjoint. Condition (b): $\lVert\Omega\rVert=1$ and $L_{p}\Omega=\widehat{p}$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum, so $\mathcal{A}\Omega=\{\widehat{p}:\ p\in\mathcal{P}_{d}\}$, which is the image of the canonical map of the Hilbert completion $\mathcal{H}$ (The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns and The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes) and hence dense in $\mathcal{H}$ by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §dense. Condition (c): for $p,q\in\mathcal{P}_{d}$, by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §algebra, Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum and condition (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state,

So $(\mathcal{H},\mathcal{A},\Omega)$ is a cyclic tracial operator algebra.

By Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §conjugation, $J$ is additive, conjugate-homogeneous, involutive and satisfies $\langle J\xi,J\eta\rangle=\langle\eta,\xi\rangle$, so it is a conjugation of $\mathcal{H}$ in the sense of Conjugation of a Complex Hilbert Space §conjugation; and for $p\in\mathcal{P}_{d}$, $J(L_{p}\Omega)=J\widehat{p}=\widehat{p^{*}}=L_{p^{*}}\Omega=L_{p}^{*}\Omega$. By the uniqueness in The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §conjugation, $J$ is the conjugation of $(\mathcal{H},\mathcal{A},\Omega)$. Finally $J\mathcal{A}J=\{JL_{p}J:\ p\in\mathcal{P}_{d}\}=\{R_{p^{*}}:\ p\in\mathcal{P}_{d}\}$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §conjugation, and this equals $\{R_{q}:\ q\in\mathcal{P}_{d}\}$ because every $q\in\mathcal{P}_{d}$ is $p^{*}$ for $p=q^{*}$, by $(q^{*})^{*}=q$ in Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint.

Claim 2. By The Tracial Algebra of a Noncommutative Law and Its Trace §algebra and The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant, $\mathcal{M}_{\lambda}$ is the commutant of $\{R_{q}:\ q\in\mathcal{P}_{d}\}=J\mathcal{A}J$, i.e. $\mathcal{M}_{\lambda}=(J\mathcal{A}J)'$. This is the algebra $M$ of The Tracial W*-Closure of a Cyclic Tracial Operator Algebra: the Commutant of the Right Action is a Tracial W*-Probability Space Equal to the Double Commutant for the cyclic tracial operator algebra $(\mathcal{H},\mathcal{A},\Omega)$ with conjugation $J$ (Claim 1). By The Tracial W*-Closure of a Cyclic Tracial Operator Algebra: the Commutant of the Right Action is a Tracial W*-Probability Space Equal to the Double Commutant §double-commutant, $\mathcal{M}_{\lambda}=\mathcal{A}''$; by The Tracial W*-Closure of a Cyclic Tracial Operator Algebra: the Commutant of the Right Action is a Tracial W*-Probability Space Equal to the Double Commutant §w-star, $(\mathcal{H},\mathcal{M}_{\lambda},\Omega)$ is a tracial W*-probability space whose conjugation is $J$. Its trace, by Tracial W*-Probability Spaces §trace and Cyclic Tracial Operator Algebras and Their Traces §trace, is $T\mapsto\langle\Omega,T\Omega\rangle$ on $\mathcal{M}_{\lambda}$, which is $\tau_{\lambda}$ by The Tracial Algebra of a Noncommutative Law and Its Trace §trace.

Claim 3. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §gns with $n=d$, $p(L_{x})=p(L_{x_{1}},\dots,L_{x_{d}})=L_{p}$ for every $p\in\mathcal{P}_{d}$. Hence $\langle\Omega,p(L_{x})\Omega\rangle=\langle\Omega,L_{p}\Omega\rangle=\lambda(p)$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum.