2014/11/30 by Martino Lupini, Lupini, Martino
Mathematics · #03C30 (Secondary) #46L07 (Primary) #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Logic (math.LO) #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1412.0281
openalex publication_date 2014/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
By means of Fraïssé theory for metric structures developed by Ben Yaacov, we show that there exists a separable 1-exact operator system \mathbbGS---which we call the Gurarij operator system---of almost universal disposition. This means that whenever E⊂ F are finite-dimensional 1-exact operator systems, ϕ:E→ \mathbbGS is a unital complete isometry, and ε >0, there is a linear extension \widehatϕ:F→ \mathbbGS of ϕ such that ||\widehatϕ||cb||\widehatϕ-1||cb≤ 1+ε . Such an operator system is unique up to complete order isomorphism. Furthermore it is nuclear, homogeneous, and any separable 1-exact operator system admits a complete order embedding into \mathbbGS. The space \mathbbGS can be regarded as the operator system analog of the Gurarij operator space \mathbbNG introduced by Oikhberg, which is in turn a canonical operator space structure on the Gurarij Banach space. We also show that the canonical ∗ -homomorphism from the universal C*-algebra of \mathbbGS to the C*-envelope of \mathbbGS is a ∗ -isomorphism. This implies that \mathbbGS does not admit any complete order embedding into a unital exact C*-algebra. In particular \mathbbGS is not completely order isomorphic to a unital C*-algebra. With similar methods we show that the Gurarij operator space \mathbbNG does not admit any completely isometric embedding into an exact C*-algebra, and in particular \mathbbNG is not completely isometric to a C*-algebra. This answers a question of Timur Oikhberg.