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

In the setting of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation, tracial W*-probability spaces $(H,M,\Omega)$ and $(K,N,\Psi)$ and the conjugation $J$ of $(H,M,\Omega)$ are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §spaces, and self-adjoint tuples $s$ in $M$, their vacuum tuples $s\Omega$, $L^{2}$ tuples, pairs $(X,Y)$ and laws $\mathrm{law}(X,Y)$ are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples; $J_{N}$ is the conjugation of $(K,N,\Psi)$, and $\lVert\cdot\rVert_{\mathrm{op}}$ is the operator norm; norms of vectors are those of $H$ or of $K$.

1. (Commutator)¶ For a self-adjoint $a\in M$ and $\zeta\in H$ we write

$$[\zeta,a]=JaJ\zeta-a\zeta\in H,$$

and likewise $[\zeta',b]=J_{N}bJ_{N}\zeta'-b\zeta'\in K$ for a self-adjoint $b\in N$ and $\zeta'\in K$.

2. (Linearity)¶ For every self-adjoint $a\in M$, all $\zeta,\eta\in H$ and every complex $c$, $[\zeta+c\eta,a]=[\zeta,a]+c[\eta,a]$.

3. (Bound)¶ For every self-adjoint $a\in M$ and every $\zeta\in H$, $\lVert[\zeta,a]\rVert\le2\lVert a\rVert_{\mathrm{op}}\lVert\zeta\rVert$.

4. (Vacuum vectors)¶ For every self-adjoint $a\in M$ and every $b\in M$, $[b\Omega,a]=(ba-ab)\Omega$.

5. (Dependence on the law)¶ Let $d\in\mathbb{N}$, let $s$ be a self-adjoint $d$-tuple in $M$ and $t$ a self-adjoint $d$-tuple in $N$, and let $P$ be an $L^{2}$ $d$-tuple of $(H,M,\Omega)$ and $Q$ an $L^{2}$ $d$-tuple of $(K,N,\Psi)$. The vacuum tuples $s\Omega$ and $t\Psi$ are $L^{2}$ $d$-tuples by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint, so $(s\Omega,P)$ and $(t\Psi,Q)$ are $L^{2}$ $2d$-tuples. Assume $\mathrm{law}(s\Omega,P)=\mathrm{law}(t\Psi,Q)$. Then

$$\sum_{j=1}^{d}\lVert[P_{j},s_{j}]\rVert^{2}=\sum_{j=1}^{d}\lVert[Q_{j},t_{j}]\rVert^{2}.$$