9
$\begingroup$

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.

$\endgroup$
0
10
$\begingroup$

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.

$\endgroup$
2
  • 4
    $\begingroup$ 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 at 16:12
  • 5
    $\begingroup$ @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 at 0:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.