  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.
