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

We analyse a notion of $C^{\ast }$-independence for ${\mathbb{Z}}_2$-graded $C^{\ast }$-algebras. We provide other notions of statistical independence for ${\mathbb{Z}}_2$-graded von Neumann algebras and prove some relationships between them. We provide a characterization for the graded nuclearity property.