On $C^*$-norms on $\mathbb{Z}_2$-graded tensor products

We systematically investigate $C^*$-norms on the algebraic graded product of $\mathbb{Z}_2$-graded $C^*$-algebras. This requires to single out the notion of a compatible norm, that is a norm with respect to which the product grading is bounded. We then focus on the spatial norm proving that it is minimal among all compatible $C^*$-norms. To this end, we first show that commutative $\mathbb{Z}_2$-graded $C^*$-algebras enjoy a nuclearity property in the category of graded $C^*$-algebras. In addition, we provide a characterization of the extreme even states of a given graded $C^*$-algebra in terms of their restriction to its even part.