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.
- 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.