On definability of C*-tensor norms

Abstract

We initiate the study of definability (in the model-theoretic sense) of C*-tensor norms. We show that neither the minimal nor maximal tensor norms are definable uniformly over all C*-algebras. The proof in the case of the minimal tensor product norm uses a deep theorem of Kirchberg characterizing exactness in terms of tensor products with matrix ultraproducts while the case of maximal tensor products uses Pisier's recent characterization of the lifting property in terms of maximal tensor products and ultraproducts. We also study the question of when one of these tensor products can be definable in a particular C*-algebra. We establish some negative results along these lines for particular C*-algebras and when the definability condition is strengthened to be computable and of a restricted quantifier-complexity; these results use the quantum complexity results MIP$^*$=RE and MIP$^{co}=$coRE. As a byproduct of our arguments, we answer a question of Fritz, Netzer, and Thom by showing that the norm on $C^*(\mathbb{F}_n\times \mathbb{F}_n)$ is not computable for any $n\in \{2,3,\ldots,\infty\}$.