Let $(A,\varphi)$ be a unital noncommutative probability space, and let $A_1,A_2 \subset A$ be tensor independent unital subalgebras. Let $B \subset A$ denote the unital subalgebra generated by $A_1 \cup A_2$, and identify $B$ with the algebraic [tensor product](/page/Tensor%20Product) $A_1 \otimes A_2$ through the multiplication map, so that $a \in A_1$ corresponds to $a \otimes 1$ and $b \in A_2$ corresponds to $1 \otimes b$. Then for every $n \in \mathbb{N}$, every $a_1,\dots,a_n \in A_1$, and every $b_1,\dots,b_n \in A_2$,

\begin{align*} \varphi(a_1b_1a_2b_2\cdots a_nb_n)=\varphi(a_1a_2\cdots a_n)\varphi(b_1b_2\cdots b_n). \end{align*}