# Reference request: Brown Ozawa and strong completely positive approximation property?

The notion of a $$C^*$$-algebra being nuclear has many equivalent characterisations. These are considered in the excellent, modern textbook $$C^*$$-Algebras and Finite-Dimensional Approximations by Brown and Ozawa. They take as definition that the identity on the $$C^*$$-algebra $$A$$ can be point-norm approximated by ccp maps which factor through matrix algebras.

The original definition is that there is only one $$C^*$$-norm on $$A\odot B$$, for any $$B$$. In the history of the subject, Kirchberg and Choi, Effros independently showed that $$A$$ is nuclear if the identity on $$A$$ can be point-norm approximated by finite-rank ccp maps from $$A$$ to itself. Kirchberg's paper calls this the strong completely positive approximation property (SCPAP) but I don't think this is common terminology now.

My questions:

• Is this result in the book by Brown and Ozawa? (Edit: By "this result" I mean precisely: that the SCPAP implies nuclearity.) I cannot seem to find it, even though it would nicely motivate the CBAP, a weaker approximation property.
• Is there any direct way to get between these two definitions? Both papers which I cite take quite a long way around, going through work of Lance and tensor products.

My motivation is to try to give a nice, expositionary, motivation of the CBAP from nuclearity; it would be nice to point to a book for this.

• @Matthew Daws: There is indeed a bridge connecting two definitions, which is quite robust and works in more general setting. It's the theory of $\delta$-norm. See Section 12 (Theorem 12.7 in particular) in Pisier's Operator Spaces book. Jan 22, 2021 at 0:32