Journal of the
Korean Mathematical Society JKMS

For independent $\sigma $-algebras $\mathcal{A}$ and $\mathcal{B}$ on $X$, $L^2(X, \mathcal{A} \vee \mathcal{B}),$ $L^2(X \times X, \mathcal{A} \times \mathcal{B}),$ and the Hilbert space tensor product $L^2(X, \mathcal{A}) \otimes L^2(X, \mathcal{B})$ are isomorphic. In this note, we show that various Hilbert $C^{*}$-algebra tensor products provide the analogous roles when independence is weakened to conditional independence.

Keywords: $L^\infty$ tensor-products, composition operators

MSC numbers: 46L06, 47C15, 47C33