Nuclear maps

Baby NCG seminar, James Gabe.

Comment: Some of you asked to see a proof of the fact, that a C*-algebra is nuclear if and only if its identity map is nuclear. This is what we will do!

Abstract: A map between C*-algebras is called nuclear if it can be approximated point-norm by maps factoring, by completely positive (cp) maps, through matrix algebras. We will show that a cp map $ V : A \to B $ is nuclear if and only if for any C*-algebra C, the induced map on the maximal tensor products $ V \otimes id_C : A \otimes_{max} C \to B \otimes_{max} C $ factors through the minimal (spatial) tensor product A \otimes_{min} C.

It follows immediately that a C*-algebra A is nuclear (i.e. A \otimes_{max} C = A \otimes_{min} C for any C) if and only if the identity map id_A is nuclear.