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

Abstract

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.