Uniqueness, universality, and homogeneity of the noncommutative Gurarij space

We realize the noncommutative Gurarij space NG defined by Oikhberg as the Fraïssé limit of the class of finite-dimensional 1-exact operator spaces. As a consequence we deduce that the noncommutative Gurarij space is unique up to completely isometric isomorphism, homogeneous, and universal among separable 1-exact operator spaces. We also prove that NG is the unique separable nuclear operator space with the property that the canonical triple morphism from the universal TRO to the triple envelope is an isomorphism. We deduce from this fact that NG does not embed completely isometrically into an exact C*-algebra, and it is not completely isometrically isomorphic to a C*-algebra or to a TRO. We also provide a canonical construction of NG, which shows that the group of surjective complete isometries of NG is universal among Polish groups. Analog results are proved in the commutative setting and, more generally, for Mn-spaces. In particular, we provide a new characterization and canonical construction of the Gurarij Banach space.