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.


1 Answer 1


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.

  • 5
    $\begingroup$ Great! That's exactly the sort of clean statement I wanted. (Sadly I think, on a quick read, that the proof is still involved. Probably that is unavoidable.) $\endgroup$ Jan 21, 2021 at 16:12
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    $\begingroup$ @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. $\endgroup$ Jan 22, 2021 at 0:32

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