56 2. Nuclear and Exact C* -Algebras
of projections PD = 1 ® qn, where the index set is the set of all finite subsets
n c 1i partially ordered by inclusion, converges to the identity operator in
the strong operator topology. On the other hand, each of the corners
PnA ® IIB(1i)pnis a nuclear C*-algebra (being isomorphic to Mn(A), where n is the dimen-
sion of the span of n) and hence the lemma above implies that A® IIB(H) is
semidiscrete, as desired.
The proposition then follows from the structure theorem for general
type I von Neumann algebras, together with the fact that a direct product
of semidiscrete von Neumann algebras is again semidiscrete. (This latter
fact is a nice exercise, using the lemma above.) D
Not wishing to dwell on classical material, we define our way to the
result we're after.
Definition 2. 7.3. A C*-algebra is type I if its double dual is a type I von
Neumann algebra.
The standard warning that type Ivon Neumann algebras are not type I
as C-algebras is probably in order. Indeed, we are about to show that type
I C-algebras are always nuclear and we saw in Proposition 2.4.9 that von
Neumann algebras rarely enjoy this property.
Proposition 2. 7.4. Type I C* -algebras are nuclear.
Proof. If A is type I, then A is a type I von Neumann algebra. By
Proposition 2.7.2, A is semidiscrete and hence Proposition 2.3.8 implies
that A is nuclear. D
Remark 2. 7.5. For those who think of type I C-algebras in terms of com-
position series, we note that this structure doesn't yield a simple proof of
nuclearity. Indeed, one would still need to know that extensions of ~uclear
algebras are nuclear. It is possible to give a C-proof of this fact, but it
is no simpler or more enlightening than the proof given above (in our not-
completely-objective opinion).
Having defined our way out of a discussion of type I C* -algebras, it is
only proper that giving examples becomes difficult.^20
Definition 2. 7.6. A is called subhomogeneous if there exists n E N such
that every irreducible representation of A is on a Hilbert space of dimension
less than or equal to n.
20For a comprehensive treatment of type I C*-algebras see [142].