I will try to put into context several well-known and one or two new facts around completely positive approximations of C*-algebras. In particular, I will explain how a separable nuclear C*-algebra can be described as an inductive limit of finite dimensional operator systems in a sense generalising the notion of NF systems of Blackadar and Kirchberg.

The setup fits in nicely with the one-sided CPAP of Ozawaand of Sato, and with Kirchberg’s subquotients theorem, thus characterising nuclearity. Our concept allows to read of pertinent properties of the algebra, such as tracial information or K-theory data, from the inductive system. This is joint work in progress with Kristin Courtney.

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