On Tsirelson pairs of C*-algebras

We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal tensor product of the pair and the space of quantum strategies obtained by using states on the maximal tensor product of the pair coincide. We exhibit a number of examples of such pairs that are ``nontrivial'' in the sense that the minimal tensor product and the maximal tensor product of the pair are not isomorphic. For example, we prove that any pair containing a C*-algebra with Kirchberg's QWEP property is a Tsirelson pair. We then introduce the notion of a C*-algebra with the Tsirelson property (TP) and establish a number of closure properties for this class. We also show that the class of C*-algebras with the TP form an axiomatizable class (in the sense of model theory), but that this class admits no ``effective'' axiomatization.