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

Abstract

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.