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arXiv · 2112.03988

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

Vitonofrio Crismale·Stefano Rossi·Paola Zurlo
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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.

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Vitonofrio Crismale, Stefano Rossi, Paola Zurlo. 2021-12-07. On $C^*$-norms on $\mathbb{Z}_2$-graded tensor products. https://arxiv.org/abs/2112.03988

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