2.3. Nuclear and exact C*-algebras 37Exercise 2.3.8 (Separable versus nonseparable). Show that a C-algebra
is exact if and only if all of its separable subalgebras· are exact. The nuclear
case is trickier but uses a technique which can be useful in other contexts:
A is nuclear if and only if each separable subalgebra B c A is contained in
a separable nuclear subalgebra C C A. (Hint for the "only if' part of the
nuclear case: If B c A is a separable subalgebra and a finite set i c B
and c: > 0 are given, then we can find c.c.p. maps cp: A -+ Mn(C) and
1/;: Mn(q-+ A such that llx - 'lj; o cp(x)ll < c: for all x E i. Letting B1 be
the C-algebra generated by B and 'lj; o cp(A), we have that B 1 satisfies a
local form of nuclearity on the set i and it's still separable. Hence we can
take a larger finite set from B1, a smaller c: > 0 and repeat the procedure.
If done carefully, the norm closure of an increasing sequence of subalgebras
constructed this way will be nuclear.)
One of the great advantages (and disadvantages) of the class of exact
C* -algebras is that they are defined via an external approximation property
(i.e., the approximating maps take values outside the algebra). This is
a disadvantage when one wants to study their fine structure. However,
external approximation has the advantage of being easier to verify in some
situations. We already saw one example of this in Exercise 2.3.2; Exercise
2.3.10 below gives another example. We need a very useful preliminary fact.
Exercise 2.3.9 (Independence ofrepresentation). Assume that Ac JB)(1i) is
a concretely represented exact C*-algebra. Show that there exist c.c.p. maps
i.pn: A-+ Mk(n)(CC) and 1/;n: Mk(n)(CC)-+ JB)(1i) such that lla-1/;noi.pn(a)ll-+
0 for all a EA. (Hint: Arveson's Extension Theorem.)
Exercise 2.3.10 (Externally locally exact). Assume A C JB)(1i) is a con-
cretely represented C -algebra which is "externally locally exact" in the
sense that for each finite set i c A and c: > 0 there exists an exact C -
algebra B c JB)(H) which almost contains i, within c: in norm. Then A is
also exact.
The exercise above fails badly for nuclear C -algebras since subalgebras
of nuclear C-algebras need not be nuclear (see Remark 4.4.4 and Corollary
9.4.5).
Exercise 2.3.11. Assume i.pn: A-+ An and 1/;n: An -+ A are c.c.p. maps
such that 1/;n o 1Pn -+ idA in the point-norm topology. Prove that .if each An
is nuclear, then so is A.
Exercise 2.3.12. Assume A c JB)(1i) and there exist c.c.p. maps 1Pn: A -+
An and 1/;n: An -+ JB)(1i) such that 1/;no1Pn-+ idA in the point-norm topology.
Use Arveson's Extension Theorem to show that A is exact if each of the An's
is exact.