Proof of The Commutator of a Square-Integrable Vector with a Bounded Self-Adjoint Operator: Linearity, the Norm Bound, Vacuum Vectors and Dependence on the Law Only

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

Claims 2 to 4 are proved for an arbitrary tracial W*-probability space $(H,M,\Omega)$ with conjugation $J$; they therefore also hold for $(K,N,\Psi)$ with $J_{N}$. Elements of $M$ lie in $\mathcal{L}(H)$ by Cyclic Tracial Operator Algebras and Their Traces §triple, and $M$ contains $I$ and is closed under sums, scalar multiples, composites and adjoints, by Cyclic Tracial Operator Algebras and Their Traces §star-algebra; also $\lVert\Omega\rVert=1$ by Cyclic Tracial Operator Algebras and Their Traces §cyclic. For $T\in\mathcal{L}(H)$ and $v\in H$ we use $\lVert Tv\rVert\le\lVert T\rVert_{\mathrm{op}}\lVert v\rVert$, which holds because $\lVert T\rVert_{\mathrm{op}}$ is a bound for $T$ 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 the triangle inequality of the norm from The Induced Norm is a Norm, and Induces a Metric.

Claim 2 (Linearity). By Conjugation of a Complex Hilbert Space §conjugation, $J$ is additive with $J(c\xi)=\overline{c}J\xi$. Since $a$ is linear,

$$JaJ(\zeta+c\eta)=J\bigl(aJ\zeta+\overline{c}\,aJ\eta\bigr)=JaJ\zeta+c\,JaJ\eta ,$$

so $JaJ$ is linear, and subtracting $a(\zeta+c\eta)=a\zeta+ca\eta$ gives $[\zeta+c\eta,a]=[\zeta,a]+c[\eta,a]$.

Claim 3 (Bound). By the same clause of Conjugation of a Complex Hilbert Space §conjugation, $\lVert J\xi\rVert^{2}=\langle J\xi,J\xi\rangle=\langle\xi,\xi\rangle=\lVert\xi\rVert^{2}$ for every $\xi\in H$. Hence $\lVert JaJ\zeta\rVert=\lVert aJ\zeta\rVert\le\lVert a\rVert_{\mathrm{op}}\lVert J\zeta\rVert=\lVert a\rVert_{\mathrm{op}}\lVert\zeta\rVert$ and $\lVert a\zeta\rVert\le\lVert a\rVert_{\mathrm{op}}\lVert\zeta\rVert$, and the triangle inequality gives $\lVert[\zeta,a]\rVert\le2\lVert a\rVert_{\mathrm{op}}\lVert\zeta\rVert$.

Claim 4 (Vacuum vectors). By The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §right-action with $S=a$ and $T=b$, $JaJ(b\Omega)=ba^{*}\Omega=ba\Omega$, as $a^{*}=a$. So $[b\Omega,a]=ba\Omega-ab\Omega=(ba-ab)\Omega$.

Claim 5 (Dependence on the law). Put $Z=(s\Omega,P)$ and $Z'=(t\Psi,Q)$, $L^{2}$ $2d$-tuples with $Z_{j}=s_{j}\Omega$ and $Z_{d+j}=P_{j}$ for $j\in[d]$, and likewise for $Z'$. Let $r=\mathbf{R}(Z)$ and $r'=\mathbf{R}(Z')$ be the resolvent transforms. By The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §consistent, $r$ is a self-adjoint $4d$-tuple in $M$ and $r'$ one in $N$, all entries of operator norm at most $1$. 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, the hypothesis $\mathrm{law}(Z)=\mathrm{law}(Z')$ gives $\lambda_{r}=\lambda_{r'}$. Comparing the formulas of The Resolvent of a Self-Adjoint Vector and the Resolvent Transform of a Square-Integrable Tuple §transform for the tuple $Z$ and for the $1$-tuple $(Z_{k})$, $(r_{2k-1},r_{2k})=\mathbf{R}(Z_{k})$ for every $k\in[2d]$, and likewise in $N$.

Fix a sequence $(q_{n})$ in $\mathcal{P}_{2,\mathrm{sa}}$ as in The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §recovery (named $(P_{n})$ there). For $k\in[2d]$ let $e_{k}=(x_{2k-1},x_{2k})$, a $2$-tuple in $\mathcal{P}_{4d}$, and let $\sigma_{e_{k}}:\mathcal{P}_{2}\to\mathcal{P}_{4d}$ be its substitution. For $j\in[d]$ and $n,m\in\mathbb{N}$ put $A_{j,n}=\sigma_{e_{d+j}}(q_{n})$, $B_{j,m}=\sigma_{e_{j}}(q_{m})$ and

$$G_{j,n,m}=A_{j,n}B_{j,m}-B_{j,m}A_{j,n}\in\mathcal{P}_{4d};$$

these polynomials do not depend on the spaces. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $A_{j,n}(r)=q_{n}(r_{2d+2j-1},r_{2d+2j})=q_{n}(\mathbf{R}(P_{j}))=:p_{j,n}$ and $B_{j,m}(r)=q_{m}(\mathbf{R}(s_{j}\Omega))=:c_{j,m}$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport, applied to the set $M$ and the inclusion map of $M$ into $\mathcal{L}(H)$, both lie in $M$. By linearity of evaluation and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism, $G_{j,n,m}(r)=p_{j,n}c_{j,m}-c_{j,m}p_{j,n}$. The same holds in $N$ with $r'$, $p'_{j,n}=q_{n}(\mathbf{R}(Q_{j}))$ and $c'_{j,m}=q_{m}(\mathbf{R}(t_{j}\Psi))$.

Fix $j\in[d]$.

Step (i). $P_{j}\in H_{\mathrm{sa}}$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples, and $s_{j}\Omega\in H_{\mathrm{sa}}$ by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint; so by The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §recovery, $p_{j,n}\Omega\to P_{j}$ as $n\to\infty$ and $c_{j,m}\Omega\to s_{j}\Omega$ as $m\to\infty$. By Claims 2 and 3 and the triangle inequality,

$$\bigl|\lVert[P_{j},s_{j}]\rVert-\lVert[p_{j,n}\Omega,s_{j}]\rVert\bigr|\le\lVert[P_{j}-p_{j,n}\Omega,s_{j}]\rVert\le2\lVert s_{j}\rVert_{\mathrm{op}}\lVert P_{j}-p_{j,n}\Omega\rVert\to0 ,$$

so $\lVert[P_{j},s_{j}]\rVert=\lim_{n}\lVert[p_{j,n}\Omega,s_{j}]\rVert$.

Step (ii). Fix $n$ and write $p=p_{j,n}$, $c=c_{j,m}$. By Claim 4, $[p\Omega,s_{j}]=(ps_{j}-s_{j}p)\Omega$, and

$$(ps_{j}-s_{j}p)\Omega-(pc-cp)\Omega=p(s_{j}-c)\Omega-(s_{j}-c)p\Omega .$$

Since $p,s_{j}-c\in M$, the two inequalities of Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the L^2 Lipschitz Bound, and Extension to Self-Adjoint Vectors §products bound the norm of the right side by $2\lVert p\rVert_{\mathrm{op}}\lVert s_{j}\Omega-c\Omega\rVert$, which tends to $0$ as $m\to\infty$ by Step (i). Hence $\lVert[p_{j,n}\Omega,s_{j}]\rVert=\lim_{m}\lVert G_{j,n,m}(r)\Omega\rVert$.

Step (iii). By The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §state, applied with the tuple $r$, bound $1$ and the unit vector $\Omega$, and since its map $\lambda_{T}$ is then the law $\lambda_{r}$, we get $\lVert G(r)\Omega\rVert^{2}=\lambda_{r}(G^{*}G)$ for every $G\in\mathcal{P}_{4d}$, and likewise $\lVert G(r')\Psi\rVert^{2}=\lambda_{r'}(G^{*}G)$. As $\lambda_{r}=\lambda_{r'}$ and norms are nonnegative, $g_{j,n,m}:=\lVert G_{j,n,m}(r)\Omega\rVert=\lVert G_{j,n,m}(r')\Psi\rVert$.

Step (iv). Steps (i) and (ii), with the order of limits: first $m\to\infty$ for each fixed $n$, then $n\to\infty$, give $\lVert[P_{j},s_{j}]\rVert=\lim_{n}\lim_{m}g_{j,n,m}$. The same steps in $(K,N,\Psi)$, using that $Q_{j}$ and $t_{j}\Psi$ are fixed by $J_{N}$ by the same two results, give $\lVert[Q_{j},t_{j}]\rVert=\lim_{n}\lim_{m}g_{j,n,m}$ with the same numbers $g_{j,n,m}$. Hence $\lVert[P_{j},s_{j}]\rVert=\lVert[Q_{j},t_{j}]\rVert$ for every $j\in[d]$, and summing the squares over $j$ proves the claim.