1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

316 11. Simple C* -Algebras

for all n. Let dn be the dimension of Sn. With a little work, one can find
natural numbers PI < q1 < p2 < q2 < p3 < q3 < · · · such that for
qm
am= p~: o er: A ---t Am= IT Ms(i)(C),
i=pm
one has
(1) (1-l/m)llxll < llam(x)ll < (1+1/m)llxll for all x E Sm and
(2) llam(xy) - am(x)am(Y)ll < d;,^4 for all x,y E Sm in the unit ball.
Note that each am is injective on Sm and hence am(Sm) is a dm-dimensional
subspace of Am. Using the Hahn-Banach Theorem, we can construct a self-
adjoint projection Pm : Am ---t am (Sm) of norm at most dm (cf. Lemma
B.10). Now define self-adjoint maps /3m: Am ---t Sm CA by
/3m = a-;;.,,^1 o Pm.

The generalized inductive system we are after is given by the algebras

Am together with the connecting maps 'Pn,m =an o f3m· (The compatibility

condition is satisfied since /3j o aj lsj = idsr) Note that

ll'Pn,m(x)ll ~ 2ll/3m(x)ll ~ 4dmllxll

for x E Am and all n > m; hence the first condition in Definition 11.1.l is

satisfied.

For the second condition we fix k, some elements x, y E Ak from the unit

ball, and E > 0. Then, for n > m > k, we have that

and that

II 'Pn,m ( 'Pm,k (x )'Pm,k(Y)) - 'Pn,m (am (/3k(x )/3k(Y))) II

~ 4dmllam(f3k(x) )am(f3k(Y )) - am (/3k(x )f3k(Y)) II

< _ 4d m (2d k )^2 d-m^4

< - l6d-m^1

ll'Pn,m(am(/3k(x)f3k(Y))) - 'Pn,k(x)r.pn,k(Y)ll

= /Ian (/3k(X )/3k(Y)) - an(f3k(x) )an(/3k(Y)) II

< - (2d k )^2 d-n^4

~ 4d-;;.^1.

It follows that

which implies we have a generalized inductive system.