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

TL;DR

This paper explores $C^*$-norms on $Z_2$-graded tensor products, establishing the minimality of the spatial norm and analyzing properties like nuclearity and states in graded $C^*$-algebras.

Contribution

It introduces the notion of compatible norms on graded tensor products and proves the spatial norm's minimality, along with nuclearity and state characterization results.

Findings

The spatial norm is minimal among compatible $C^*$-norms.

Commutative $Z_2$-graded $C^*$-algebras are nuclear in the graded category.

Extreme even states are characterized by their restriction to the even part.

Abstract

We systematically investigate $C^*$-norms on the algebraic graded product of $\mathbb{Z}_2$-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 $\mathbb{Z}_2$-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.