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We give a characterization of $C^*$-independence for $\mathbb{Z}_2$-graded$C^*$-subalgebras of $\mathcal{B}(\mathcal{H})$ based on the Schliederproperty. We provide other notions of statistical independence for$\mathbb{Z}_2$-graded von Neumann algebras and prove some relationships betweenthem. We provide a characterization for the graded nuclearity property.

$C^*$-independence for $\mathbb{Z}_2$-graded $C^*$-algebras

$C^$-independence for $\mathbb{Z}_2$-graded $C^$-algebras