Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X_1,\dots,X_d\in L^\infty(\Omega,\mathcal F,\mathbb P)$ be bounded complex-valued random variables. Let $H:=L^2(\Omega,\mathcal F,\mathbb P)$, with [inner product](/page/Inner%20Product)
For each $i\in\{1,\dots,d\}$, define the multiplication operator $M_i:H\to H$ by $M_i f:=X_i f$. Let $\Phi:\mathcal L(H)\to\mathbb C$ be the vector state
where $1_\Omega$ is the constant-one element of $H$. Then $M_i^*=M_{\overline{X_i}}$ for every $i$, and for every integer $k\geq 1$, every choice of indices $i_1,\dots,i_k\in\{1,\dots,d\}$, and every choice of exponents $\varepsilon_1,\dots,\varepsilon_k\in\{1,*\}$,