1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2. 
   
   
   
   7. Type I C* -algebras 57
   
   
   
   

Proposition 2.7.7. If A is subhomogeneous, then A is type I (hence nu-

clear).

Proof. For each natural number j let Ij be the set of pure states with

j-dimensional GNS representations and let

'Trj: A --t II Mj(C)

iEij

be the direct sum of all the corresponding GNS maps. There is a natural

isomorphism^21

II Mj(C) ~ Mj(£^00 (Ij))

iEij

and hence we get an embedding

for some sufficiently large k, since the direct sum of all irreducible repre-
sentations is always faithful. Hence the double dual of A is isomorphic to a
subalgebra of

( £^00 (11) EB ... EB Mk(£^00 (h))) ~ £^00 (11) EB ... EB Mk(£^00 (Ik)**).

Since exactness passes to subalgebras, Proposition 2.4.9 tells us that the
double dual of A is also a von Neumann algebra of type I. D

Exercises

Exercise 2.7.1 (Semidiscreteness and direct sums). Show that if Mi, i EI,
is a collection of semidiscrete von Neumann algebras, then the direct sum

is also semidiscrete.

Exercise 2. 7.2 (Semidiscreteness and inductive limits). It is extremely easy
to show that increasing unions of nuclear or exact C* -algebras are again
nuclear or, respectively, exact (Exercise 2.3.7). Try to prove that if M has
an increasing sequence of semidiscrete subalgebras M1 C M2 C · · · whose
union is ultra weakly dense in M, then M is also semi discrete.^22

21This well-known fact makes a nice exercise. Or, refer to Lemma 3.9.4.

(^22) If you succeed, without quoting difficult theorems, then you should publish the proof! It
turns out that M will be semidiscrete in this case but the point of the exercise is that the obvious
adaptation of the analogous C*-result fails, and it is instructive to see where the argument breaks
down.