Uniqueness, universality, and homogeneity of the noncommutative Gurarij space (Q291758)

This paper studies certain Fraïssé classes and their Fraïssé limits. Fraïssé limits are objects strongly connecting combinatorics and model theory. Given a Fraïssé class, its limit is a unique object up to isomorphism carrying universality and homogeneity properties. The author shows that \textit{T. Oikhberg}'s noncommutative generalization of the Gurarij space \(\mathbb N\mathbb G\) [Arch. Math. 86, No. 4, 356--364 (2006; Zbl 1119.46045)] is a Fraïssé limit for the class of separable \(1\)-exact operator spaces (\(1\)-exactness for operator spaces is the correspondent of exactness for \(C^*\)-algebras), and concludes that \(\mathbb N\mathbb G\) is the unique homogeneous object which is universal among this class of operator spaces, in parallel to Ben Yaacov's proof that the Gurarijpace \(\mathbb G\) is a Fraïssé limit. A similar result, using similar techniques involving partial isometries of operator spaces, is proved, in Section 3, for the Gurarij \(M_n\)-space \(\mathbb G_n\).NEWLINENEWLINESection 5 is dedicated to consequences. It is shown that every separable exact \(C^*\)-algebra embeds into \(\mathbb N \mathbb G\), and that the latter does not embed in a separable exact \(C^*\)-algebra. Also, model theoretical and combinatorial consequences of the fact that \(\mathbb N \mathbb G\) and \(\mathbb G_n\) are Fraïssé limits are given.