1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

308 10. Summary

Proof. Let i c A/ I be a finite set and let E > 0. Let X C A/ I be a finite-
dimensional operator system which contains i and {ab : a, b E i}. Fix a
u.c.p. splitting <p: X -----> A. Now take a quasicentral approximate unit of
projections, say {pn}, and consider the (isometric, though no longer unital)
completely positive splittings 'Pn(x) = (1-Pn)<p(x)(l - Pn)· We claim that
for sufficiently large n, these maps are E-multiplicative on i; evidently this
will imply the result.
For a, b-E i we estimate

ll'Pn(ab) _:__ 'Pn(a)<pn(b)ll

= II (1 -Pn)<p( ab) (1 - Pn) - (1 - Pn)'P( a) (1 - Pn)<p(b) (1 - Pn) II

:S 11(1-Pn) (<p(ab) - <p(a)<p(b)) (1-Pn)ll

• 11(1-Pn) ((1-Pn)<p(a) - <p(a)(l - Pn))<p(b)(l - Pn)ll·

Since <p is a splitting, Jl7r(x) II = lim II (1 -Pn)x(l - Pn) II and {pn} is quasi-

central, the result follows easily. D

Note that local liftability is automatic whenever A is locally reflexive

(e.g., exact).

Inductive limits. For inductive limits with injective connecting maps, a

simple application of Arveson's Extension Theorem shows that quasidiago-

nality passes to inductive limits. For arbitrary inductive limits this is false;

we will see a counterexample in Remark 17.3.3. However, in the presence of

exactness nothing funny happens.

Proposition 10.3.6. If A= limAn ----+ and each An is QD and locally reflexive,

then A is QD.

Proof. We may assume everything is unital. Essentially by definition, A is

a subalgebra of

IT An

ffiAn'

Hence we have a quasidiagonal extension 0 -----> EB An -----> B -----> A -----> 0.
Moreover, Bis clearly QD and the quotient mapping is locally liftable, since
each An is locally reflexive. Thus quasidiagonality passes to the quotient
A. D

Direct sums and products.· Both co and £^00 -direct sums of QD algebras
are again QD. This is easily deduced from the definition.