Proof of Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators

Each result cited below is universally quantified over the data in its own statement. Recall that $(H,M,\Omega)$ is a cyclic tracial operator algebra with $M=M''$ by Tracial W*-Probability Spaces §space, so $I\in M$ and $M$ is closed under sums, complex multiples, composition and adjoints by Cyclic Tracial Operator Algebras and Their Traces §star-algebra, and $\lVert\Omega\rVert=1$ by Cyclic Tracial Operator Algebras and Their Traces §cyclic; hence $p(T)\in M$ for every $p\in\mathcal{P}_{n}$ and every $n$-tuple $T$ in $M$, by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport applied with $\mathcal{A}=M$ and $\Phi$ the identity map. Evaluation $p\mapsto p(T)$ is linear by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation.

Step 1 (Uniqueness). If $s,s'$ are self-adjoint $d$-tuples in $M$ with $s\Omega=s'\Omega=X$, then for each $j\in[d]$ we have $s_{j}-s'_{j}\in M$ and $(s_{j}-s'_{j})\Omega=0$, so $s_{j}=s'_{j}$ by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §separating.

For Steps 2 to 8 fix $j\in[d]$.

Step 2 (Reference realisation). Since $\lambda\in\Sigma_{d,R}\subseteq\Sigma_{d}$ (the inclusion by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law), the triple $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ is a tracial W*-probability space by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star. Let $t=(L_{x_{1}},\dots,L_{x_{d}})$. Each $t_{k}$ lies in $\mathcal{M}_{\lambda}$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra, is self-adjoint 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 §adjoint (as $x_{k}^{*}=x_{k}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint), and satisfies $\lVert t_{k}\rVert_{\mathrm{op}}\le R$ 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 with $r=R$. By the definition of $\lambda_{t}$ and The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law, $\lambda_{t}(p)=\langle\Omega_{\lambda},p(t)\Omega_{\lambda}\rangle=\lambda(p)$ for all $p$, so $\lambda_{t}=\lambda$, and by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded $\mathrm{law}(t\Omega_{\lambda})=\kappa_{d}(\lambda)=\mathrm{law}(X)$.

Step 3 (Transfer through the resolvent law). By The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §consistent, $\mathbf{R}(X)$ is a self-adjoint $2d$-tuple in $M$ with entries of operator norm at most $1$, and $\mathbf{R}(t\Omega_{\lambda})=\mathbf{R}(t)$; by The Law of the Resolvent Transform is a Lipschitz Function of the Law, and Reconstruction of a Square-Integrable Tuple from a Bounded Tuple with the Law of Its Transform §law and Step 2, $\lambda_{\mathbf{R}(X)}=\lambda_{\mathbf{R}(t\Omega_{\lambda})}=\lambda_{\mathbf{R}(t)}$. By the definition of the transforms and its operator version, $\mathbf{R}(X_{j})=(\mathbf{R}(X)_{2j-1},\mathbf{R}(X)_{2j})$ and $\mathbf{R}(t_{j})=(\mathbf{R}(t)_{2j-1},\mathbf{R}(t)_{2j})$. Let $a=(x_{2j-1},x_{2j})$, a $2$-tuple in $\mathcal{P}_{2d}$. For every $2d$-tuple $T$ of operators, $a(T)=(T_{2j-1},T_{2j})$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, so, where $\sigma_{a}:\mathcal{P}_{2}\to\mathcal{P}_{2d}$ is the substitution of $a$, $\sigma_{a}(r)(T)=r(T_{2j-1},T_{2j})$ for $r\in\mathcal{P}_{2}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution. Evaluating the equal laws at $\sigma_{a}(r)$ gives, for every $r\in\mathcal{P}_{2}$,

$$(\ast)\qquad\langle\Omega,r(\mathbf{R}(X_{j}))\Omega\rangle=\langle\Omega_{\lambda},r(\mathbf{R}(t_{j}))\Omega_{\lambda}\rangle.$$

Step 4 (Uniform polynomials). Let $n\in\mathbb{N}$. First choose $y=1+2nR^{3}$, so $y\ge1$ and $R^{3}/y^{2}\le1/(2n)$; then put $\varepsilon=1/(2ny^{2})$; then take $p\in\mathcal{P}_{2}$ from Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §transform for $s=y$ and $\varepsilon$; finally set $q_{n}=\tfrac{y^{2}}{2}(p+p^{*})$, which lies in $\mathcal{P}_{2,\mathrm{sa}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. Let $K$ be a complex Hilbert space and $A\in\mathcal{L}(K)$ self-adjoint with $\lVert A\rVert_{\mathrm{op}}\le R$. Put $E=R_{y}(A)-p(\mathbf{R}(A))$, so $\lVert E\rVert_{\mathrm{op}}\le\varepsilon$. The entries of $\mathbf{R}(A)$ are self-adjoint by Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §parts, so $p(\mathbf{R}(A))^{*}=p^{*}(\mathbf{R}(A))$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint; with $R_{y}(A)^{*}=R_{-y}(A)$ from Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §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, $E^{*}=R_{-y}(A)-p^{*}(\mathbf{R}(A))$, and $\lVert E^{*}\rVert_{\mathrm{op}}=\lVert E\rVert_{\mathrm{op}}$ 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. Hence

$$A-q_{n}(\mathbf{R}(A))=\Bigl(A-\tfrac{y^{2}}{2}\bigl(R_{y}(A)+R_{-y}(A)\bigr)\Bigr)+\tfrac{y^{2}}{2}(E+E^{*}),$$

and by Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §recovery and the norm rules of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations, $\lVert A-q_{n}(\mathbf{R}(A))\rVert_{\mathrm{op}}\le R^{3}/y^{2}+y^{2}\varepsilon\le1/n$. The polynomial $q_{n}$ depends only on $n$ and $R$.

Step 5 (Approximants and their moments). Let $b_{n}=q_{n}(\mathbf{R}(X_{j}))\in M$ and $c_{n}=q_{n}(\mathbf{R}(t_{j}))\in\mathcal{L}(\mathcal{H}_{\lambda})$; both are self-adjoint by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint, as $\mathbf{R}(X_{j})$ and $\mathbf{R}(t_{j})$ have self-adjoint entries (Step 3 and The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §consistent). By Step 4 with $A=t_{j}$, $\lVert c_{n}-t_{j}\rVert_{\mathrm{op}}\le1/n$, so $\lVert c_{n}\rVert_{\mathrm{op}}\le R+1/n$. For $m\in\mathbb{N}$, applying $(\ast)$ to the product $r$ of $2m$ factors $q_{n}$ and using Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism, $\langle\Omega,b_{n}^{2m}\Omega\rangle=\langle\Omega_{\lambda},c_{n}^{2m}\Omega_{\lambda}\rangle$; since $c_{n}^{2m}$ is the value of the monomial $x_{1}^{2m}\in\mathcal{P}_{1}$ at the $1$-tuple $(c_{n})$ (Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism) and $\lVert\Omega_{\lambda}\rVert=1$ 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, The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §state gives $|\langle\Omega,b_{n}^{2m}\Omega\rangle|\le(R+1/n)^{2m}$.

Step 6 (A moment bound gives a right multiplication bound). Let $b\in M$ be self-adjoint and $C>0$ real with $|\langle\Omega,b^{2m}\Omega\rangle|\le C^{2m}$ for all $m\in\mathbb{N}$; we show $\lVert bT\Omega\rVert\le C\lVert T\Omega\rVert$ for every $T\in M$. Put $u_{1}=\tfrac12(T+T^{*})$ and $u_{2}=\tfrac{1}{2i}(T-T^{*})$; they lie in $M$, are self-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-calculus, and $T=u_{1}+iu_{2}$. Let $V=(b,u_{1},u_{2})$ and $\nu=\lambda_{V}$; then, with $R'=1+\lVert b\rVert_{\mathrm{op}}+\lVert u_{1}\rVert_{\mathrm{op}}+\lVert u_{2}\rVert_{\mathrm{op}}$, $\nu\in\Sigma_{3}$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law applied with the bound $R'$, so $\nu$ is a tracial state (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law). By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism, together with the linearity of evaluation, $x_{1}^{2m}(V)=b^{2m}$, $(x_{2}+ix_{3})(V)=T$ and $(x_{1}(x_{2}+ix_{3}))(V)=bT$. The number $\nu(x_{1}^{2m})=\langle\Omega,b^{2m}\Omega\rangle$ is real (as noted 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), so $\nu(x_{1}^{2m})\le C^{2m}$, and 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 with $a=x_{1}$ ($x_{1}\in\mathcal{P}_{3,\mathrm{sa}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint), $S=C$ and $p=x_{2}+ix_{3}$ gives $\lVert x_{1}p\rVert_{\nu}\le C\lVert p\rVert_{\nu}$. By The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §state (with bound $R'$), $\lVert q\rVert_{\nu}=\lVert q(V)\Omega\rVert$ for every $q\in\mathcal{P}_{3}$, whence $\lVert bT\Omega\rVert\le C\lVert T\Omega\rVert$. Applied to $b=b_{n}$, $C=R+1/n$ (Step 5) and $T=S^{*}$, and using $\lVert S^{*}\Omega\rVert=\lVert S\Omega\rVert$ from The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §trace: $\lVert b_{n}S^{*}\Omega\rVert\le(R+1/n)\lVert S\Omega\rVert$ for every $S\in M$.

Step 7 ($b_{n}\Omega\to X_{j}$). Take $(P_{m})$ from The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §recovery and put $D=q_{n}-P_{m}$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint, $(D^{*}D)(\mathbf{R}(X_{j}))=D(\mathbf{R}(X_{j}))^{*}D(\mathbf{R}(X_{j}))$, and likewise at $\mathbf{R}(t_{j})$; so $(\ast)$ with $r=D^{*}D$ and the identity $\langle v,T^{*}Tv\rangle=\lVert Tv\rVert^{2}$ of 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 give

$$\lVert b_{n}\Omega-P_{m}(\mathbf{R}(X_{j}))\Omega\rVert=\lVert c_{n}\Omega_{\lambda}-P_{m}(\mathbf{R}(t_{j}))\Omega_{\lambda}\rVert .$$

Now $X_{j}$ is fixed by $J$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples) and $t_{j}\Omega_{\lambda}$ is fixed by the conjugation of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ (Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law); and $\mathbf{R}(t_{j}\Omega_{\lambda})=\mathbf{R}(t_{j})$ by The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §consistent. So, as $m\to\infty$, $P_{m}(\mathbf{R}(X_{j}))\Omega\to X_{j}$ and $P_{m}(\mathbf{R}(t_{j}))\Omega_{\lambda}\to t_{j}\Omega_{\lambda}$ by The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §recovery, and continuity of the norm gives $\lVert b_{n}\Omega-X_{j}\rVert=\lVert(c_{n}-t_{j})\Omega_{\lambda}\rVert\le\lVert c_{n}-t_{j}\rVert_{\mathrm{op}}\le1/n$ 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 and Step 5.

Step 8 (The operator $s_{j}$). Let $S\in M$. By The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §right-action, $JSJ\in M'\subseteq\mathcal{L}(H)$ and $JSJ(b_{n}\Omega)=b_{n}S^{*}\Omega$. As $JSJ$ is bounded, Step 7 gives $JSJX_{j}=\lim_{n}b_{n}S^{*}\Omega$, so by Step 6 $\lVert JSJX_{j}\rVert\le\lim_{n}(R+1/n)\lVert S\Omega\rVert=R\lVert S\Omega\rVert$. By Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §right-bounded with $C=R$ there is $s_{j}\in M$ with $s_{j}\Omega=X_{j}$ and $\lVert s_{j}\rVert_{\mathrm{op}}\le R$. By The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §conjugation, $s_{j}^{*}\Omega=J(s_{j}\Omega)=JX_{j}=X_{j}=s_{j}\Omega$, and $s_{j}^{*}-s_{j}\in M$, so $s_{j}^{*}=s_{j}$ by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §separating. Thus $s=(s_{1},\dots,s_{d})$ is a self-adjoint $d$-tuple in $M$ with $s\Omega=X$; with Step 1 this proves clause 1, and the bound proves clause 2.

Step 9 (Law). By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded, $\kappa_{d}(\lambda_{s})=\mathrm{law}(s\Omega)=\mathrm{law}(X)=\kappa_{d}(\lambda)$, where $\lambda_{s}\in\Sigma_{d}$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law. Since $(\Sigma_{d},W_{2})$ is a metric space by The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §metric and $\kappa_{d}$ is its canonical map, $\kappa_{d}$ is injective by The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §isometry; hence $\lambda_{s}=\lambda$, which is clause 3.