The non-commutative Gurarii space

Abstract

We construct a "universal space" for 1-exact finite dimensional operator spaces (an analogue of the Gurarii space).

Operator space and operator system analogs of Kirchberg's nuclear embedding theorem

“…(Modulo uniqueness of NG, this also follows from [22,Theorem 1.1] and the proof of [8,Theorem 4.7].) Such a result can be regarded as an operator space version of Kirchberg's exact embedding theorem.…”

“…The Gurarij operator space NG has been defined in [22] to be a separable 1-exact operator space satisfying the following approximate injectivity property: whenever E ⊂ F are finite-dimensional 1-exact operator spaces, ϕ : E → NG is a complete isometry, and ε > 0, then φ can be extended to a linear map φ :…”

“…Furthermore the inclusion functor from the category of M q -spaces to the category of operator spaces determines an equivalence of categories with inverse the functor MIN q . The notion of M q -space has been introduced and studied in [18], and used in [23,22,19].…”

“…Existence of the Gurarij operator space was first proved in [22], while uniqueness was established in [19]. It is worth mentioning here that, albeit being 1-exact, NG does not admit any completely isometric embedding into an exact C*-algebra by [20,Corollary 4.16].…”

The complete isomorphism class of an operator space

“…Recall that (see [4,6,7]), for an operator space E and n ∈ N, we define an operator space MIN n (E) to be isometric to E on the Banach space level, and…”