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

Abstract

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

19 pages, to appear in Banach Journal of Mathematical Analysis