36 2. Nuclear and Exact C* -AlgebrasExercises
Exercise 2.3.1 (Finite sets and c:'s). Give the proper "local" formulations
of nuclearity and exactness. What additional ingredient needs to be localized
for semidiscreteness?
Exercise 2.3.2 (General subalgebras). It is easily seen that exactness passes
to subalgebras, but the same is not true for nuclearity. Where does the proof
break down?
Exercise 2.3.3 (Subalgebras with conditional expectations). If A is nuclear
and B c A is a C -subalgebra such that there exists a conditional expec-
tation : A -+ B, then B is also nuclear. Formulate a similar result for
semidiscreteness. In particular, prove that an arbitrary von Neumann sub-
algebra of a separable semidiscrete algebra of type II 1 is again semidiscrete.
(Hint: See Lemma 1.5.11 for the II1 case.)
Exercise 2.3.4 (Hereditary subalgebras and nuclearity). Prove that a hered-
itary subalgebra of a nuclear C-algebra is again nuclear. In particular, nu-
clearity passes to ideals. (Hint: If en is an approximate unit of a hered-
itary subalgebra B C A, then the c.c.p. maps n: A -+ B defined by
n(a) = enaen have the property that lln(b) - bll -+ 0 for all b EB.)
Exercise 2.3.5 (Unitizations). Show that a nonunital C-algebra A is nu-
clear (resp. exact) if and only if the unitization A is nuclear (resp. exact).
Exercise 2.3.6 (Direct sums). Prove that a finite direct sum Ai EEl · · · EElAk
is nuclear (resp. exact) if and only if each Ai is nuclear (resp. exact). It
turns out that the £^00 -direct sum of nuclear C-algebras need not be exact.
In fact,
IJ Mn(CC) = {(xn) : Xn E Mn(CC), sup llxnll < 00}
nEN n
is not exact.^7 What happens with the co-direct sum (i.e., sequences tending
to zero in norm)?
Exercise 2.3. 7 (Locally nuclear). Prove that a C -algebra which is "locally
nuclear" is nuclear. That is, if for each finite set ~ c A and c: > 0 one can
find a nuclear subalgebra B c A such that B almost contains ~' within c:
in norm, then A is nuclear. In particular, the class of nuclear C -algebras is
closed under taking inductive limits with injective connecting maps.^8 (Hint:
You'll need Arveson's Extension Theorem.)
(^7) This follows from the fact that there exist nonexact residually finite-dimensional C-
algebras, such as the full group C-algebra of a free group (see Theorem 7.4.1).
(^8) Actually, injectivity of the connecting maps is not necessary, but then you need to know
that nuclearity passes to quotients - a deep, hard fact. Exact C* -algebras are also closed under
arbitrary inductive limits.