1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

302 10. Summary

Extensions.

Proposition 10.1.3. If 0 ---+ I ---+ A ---+ B ---+ 0 is short exact and both I

and B are nuclear, then so is A (Exercise 3.8.1).

Quotients. At present, there is no elementary proof of the following fact.

Theorem 10.1.4. If I <lA and A is nuclear, then A/ I is also nuclear (Corol-

lary 9.4.4).

Inductive limits. For inductive limits with injective connecting maps, the
following result is easy (use Arveson's Extension Theorem). In general, it is
not. But quotients of nuclear C*-algebras are nuclear, so everything reduces
to the injective case.

Theorem 10.1.5. An inductive limit of nuclear C*-algebras is nuclear.

Direct sums and products. Infinite direct products (aka £^00 -direct sums)

of nuclear algebras are rarely nuclear; ITnEN Mk(n) (C) is nuclear if and only

if sup k(n) < oo (Proposition 2.4.9). However, for sequences tending to zero

in norm (aka co-direct sums), all is well.

Proposition 10.1.6. Let Ai, i E I, be a collection of C*-algebras. Then

EBiEJ Ai is nuclear if and only if each Ai is nuclear.

The proof is trivial - take a (finite) direct sum of c.c.p. approximations
on the Ai's.

Tensor products.

Proposition 10.1.7. Let A and B be arbitrary C*-algebras. The following
statements are equivalent:
(1) both A and B are nuclear;
(2) A® B is nuclear;
(3) A ®max B is nuclear.

Proof. We only prove (1) 9 (3) as the equivalence of (1) and (2) is similar.

First assume A and B are nuclear. Taking maximal tensor products of
c.c.p. approximations, one easily checks that A ®max Bis also nuclear. For
the other direction, choose a positive norm-one element b E B and a state 'P
on B such that 'fl(b) = 1. Then, for the c.c.p. embedding lb: A---+ A®maxB,
defined by lb (a) = a® b, and the slice map idA ® 'P: A ®max B ---+ A® C ~ A,
we have (idA®'P)olb = idA. Since A®maxB is nuclear, A is nuclear, too. D