On C*‑norms on Z_2‑graded tensor products

We systematically investigate C*-norms on the algebraic graded product of 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 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.