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We systematically investigate    $$C^*$$  C ∗ -norms on the algebraic graded product of    $${{\mathbb {Z}}}_2$$  Z 2 -graded    $$C^*$$  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    $$C^*$$  C ∗ -norms. To this end, we first show that commutative    $${{\mathbb {Z}}}_2$$  Z 2 -graded    $$C^*$$  C ∗ -algebras enjoy a nuclearity property in the category of graded    $$C^*$$  C ∗ -algebras. In addition, we provide a characterization of the extreme even states of a given graded    $$C^*$$  C ∗ -algebra in terms of their restriction to its even part.

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