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](https://www.ams.org/publications/authors/books/postpub/gsm-88) 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](https://onlinelibrary.wiley.com/doi/abs/10.1002/mana.19770760115) and [Choi, Effros](https://www.jstor.org/stable/2373876) 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.



Reference request: Brown Ozawa and strong completely positive approximation property?
I am not sure if it is in Brown and Ozawa, but it is in Pisier's recent book "Tensor Products of C\*-algebras and Operator Spaces" as Corollary 10.16. It may also be in his earlier Operator Spaces book, but my copy isn't with me.


