AbstractWe 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.
Hochschild cohomology of twisted tensor products
