104 3. Tensor Products
Theorem 3.8.7 (Choi and Effros, Kirchberg). For a C*-algebra A, the
following statements are equivalent:
(1) A is nuclear;
(2) for every C* -algebra B, there is a unique C* -norm on A 8 B.^25Proof. We already observed (1) ~ (2) - see Proposition 3.6.12; thus we
assume there is always a unique C*-norm on A 8 B. By Exercise 3.7.2 we
may assume A is unital. Applying Theorem 3.8.5 with <p = l: A ----+ A**,
we see that the inclusion of A into its double dual is a weakly nuclear map.
This implies A is nuclear (Proposition 2.3.8 and Exercise 2.3.13). D
Finally, we provide the converse to Lemma 3.6.10, as promised earlier.Corollary 3.8.8. Let t.p: A ----+ B be a u.c.p. map. Then <p is nuclear if and
only if for every C* -algebra C, the maximal tensor product map
t.p 0max ido: A 0max C----+ B 0max C
factors through A 0 C.
Proof. Since we only have to show the "if'' direction, let B c B** C IIB(1iu)
be the universal representation and assume
i.p 0max id(B**)': A @max (B**)' ----+ B @max (B**)'factors through A 0 (B)'. Composing this map with the product map
B 0max (B)'----+ IIB(1iu), we see that the product map
<p x l(B**)': A 8 (B**)' ----+ IIB(1iu)is min-continuous and hence <p is weakly nuclear as a map from A to B**.
As with the previous result, this is enough to imply nuclearity of <p.^26 D
The C-purists may find our proof of Theorem 3.8.7 somewhat annoying
since we relied so heavily on W -machinery. There is an "alternate" proof
which bypasses the W*-preliminaries and goes directly to Theorem 3.8.7.
However, it is virtually identical to the proof of Theorem 3.8.5, thus not
much easier.
Here is a sketch: Suppose A has the property that A 8 C has a unique
C-norm for every C-algebra C. Invoking Lemma 2.3.4, it suffices to show
that idA is in the point-weak closure of the factorable maps. Hence we fix a
finite set~ C A, finite set of states S C S(A) ands > 0. Now we let f be the
average of the states in S, ?r: A----+ IIB(L^2 (A, f)) be the GNS representation
(^25) 0riginally this condition was the definition of nuclearity.
(^26) Since <p takes values in B and is weakly nuclear as a map into B**, this should now be
routine.