Uniqueness, universality, and homogeneity of the noncommutative Gurarij space

Abstract

We realize the noncommutative Gurarij space $\mathbb{NG}$ defined by Oikhberg as the Fra\"ıssé limit of the class of finite-dimensional $1$-exact operator spaces. As a consequence we deduce that the concommutative Gurarij space is unique up to completely isometric isomorphism, homogeneous, and universal among separable $1$-exact operator spaces. We also prove that $\mathbb{NG}$ 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 $\mathbb{NG}$ 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 $\mathbb{NG}$, which shows that the group of surjective complete isometries of $\mathbb{NG}$ is universal among Polish groups. Analog results are proved in the commutative setting and, more generally, for $M_{n}$-spaces. In particular, we provide a new characterization and canonical construction of the Gurarij Banach space.