1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2.4. First examples 39

we have

lf(x) - f(y)I < E

for any pair of points x, y E Ui. Let Yi E Ui be arbitrarily chosen and

{0-1, ... , o-n} be a partition of unity subordinate to the cover {U1, ... , Un}·

Define cp: A -----+ en by cp(f) = (f(y1), ... , f(Yn)). Evidently 'Pis a unital

*-homomorphism, hence a u.c.p. map. We then define 'l/J: en-----+ A by

n

(d1, ... , dn) f--)-L dWi·

i=l

It is not hard to see that the map 'ljJ is positive and it is a general fact that if
either the range or domain of a positive map is an abelian C* -algebra, then
the map is automatically c.p. ([141, Theorem 3.9 and Theorem 3.11]).^9 By
Exercise 2.1.2, we are left to estimate II! - 'ljJ o ¢(!)11:
n n n
II! -'ljJ^0 ¢(!)11 = 11(2:: 0-i)f - L f(Yi)o-ill = II l::U -f(yi)l)o-ill :::; E
i=l i=l i=l

for every f E 'J.^0

Remark 2.4.3. Nuclearity is sometimes regarded as the noncommutative
analogue of having a partition of unity. Though the analogy is not perfect,
it has substance and the proof above explains this point of view.

Corollary 2.4.4. For every locally compact Hausdorff space X and natural

number n EN, Mn(Co(X)) is nuclear.

Proof. More generally, it is easily seen that if A is nuclear, then so is

~~· 0

Corollary 2.4.5. Every approximately homogeneous {AH) algebra is nu-

clear.

Proof. By definition, an AH algebra is an inductive limit, with injective
connecting maps, of algebras Ak, where each Ak is a finite direct sum of
algebras of the form

PMn(Co(X))P,

(^9) In our setting the proof is straightforward. We identify Mk(Cn) with Mk(C) E9 · · · E9 Mk(C)
(direct sum n times) and it is clear that
n
'1fJk(T1 E9 .. · E9 Tk) = 2= '.Z1 0 O'z,
Z=l
where T 0 O' E Mk(O(X)) ~ O(X,Mk(C)) is the matrix-valued function :r: f-+ O'(:r:)T. Evidently
this implies that '1f; is completely positive.