1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

304 10. Summary

We have seen that exactness is also equivalent to properties C and C'

(Theorem 9.3.1). Finally, as mentioned in the preface, a fundamental theo-

rem of Kirchberg is completely missing from these notes; at the very least,

we should state it.

Theorem 10.2.2 ([107], [168]). A separable C*-algebra is exact if and only

if it is isomorphic to a subalgebra of the Cuntz algebra 02.

Subalgebras.

Proposition 10.2.3. Exactness passes to subalgebras.

This is immediate from the definition.

Extensions. We will show in Section 13.4 that extensions of exact C* -

algebras need not be exact, in general. However, the following result is very

useful.

Theorem 10.2.4. Let 0 -+ I -+ A -+ B -+ 0 be a short exact sequence

where both I and B are exact and A is unital. Then A is exact if and only

if the extension is locally split (Exercise 3.9.8).

Quotients. At present, this is one of the hardest C* -results, ever.

Theorem 10.2.5. A quotient of an exact C* -algebra is again exact ( Corol-

lary 9.4.3).

Inductive limits. Just like the nuclear case, the following fact is trivial
when the connecting maps are injective and it is very hard in general.

Theorem 10.2.6. An inductive limit of exact C* -algebras is exact.

Direct sums and products. This is identical to the nuclear case.

Tensor products. The proof of the following result is very similar to the
nuclear case.

Proposition 10.2. 7. Let A and B be arbitrary C* -algebras. The following
are equivalent:
(1) both .A and B are exact;
(2) A 0 B is exact.

What happened to the maximal tensor product? Doesn't the nuclear
proof carry over verbatim? Seriously, think about it for a minute. What
goes wrong with the proof?