1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

112 3. Tensor Products

Exercise 3.9"6. Prove that A is exact if and only if the sequence

O, ( ~Mn(C)) Q9A, (I]Mn(C)) Q9A, (~:::~~~) Q9A,O

is exact.

Exercise 3.9" 7. Prove that A is exact if and only if the sequence

O __, JK(H) Q9 A __, Illl(H) Q9 A __, ~~~~ Q9 A __, O

is exact for some infinite-dimensional Hilbert space H.

Exercise 3.9.8. Assume 0 __, J __,A__, A/ J __, 0 is a locally split (Defini-

tion 3. 7.5) short exact sequence. Prove that if J and A/ J are exact, then

so is A.^29 (Hint: Revisit Proposition 3.7.6 and then do a big diagram chase

(aka the 3 x 3 Lemma) through

0 0 0

1 1 1

0---+ JQ9J ---+ BQ9 J ---+ B/IQ9 J ---+ 0

1 1 1

0---+ IQ9A ---+ B Q9A ---+ B/IQ9A ---+ 0

1 1 1

O ---+ I Q9 A/ J ---+ BQ9A/J ---+ B/I Q9 A/J ---+ 0

1 1 1
0 0 0
Exercise 3.9.9. If one assumes a nontrivial tensor product fact, a very
short proof of Kirchberg's Theorem can be given. Assume that there is
a unique C-norm on C(JF 00 ) 8 Illl(H) (this is a fact - see Section 13.2).
Deduce Theorem 3.9.1. (Hint: If A C Illl(H) is Q9-exact, then try to prove
that A Q9max B __, Illl(H) Q9max B factors through the spatial tensor product,
for every B.)

3.10. ·References

Takesaki's Theorem was proved in [182]; the result is quite surprising, as the
injective Banach space norm need not be the smallest, in general. Continuity
of c.p. maps on maximal tensor products relies on [9], where Arveson intro-
duced c.c. maps and proved his fundamental extension theorem. The Trick is

(^29) Kirchberg has constructed counterexamples when the sequence is not locally split (see
Remark 13.4.2). ·