1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

240 7. Quasidiagonal C* -Algebras

Proposition 7.1.9. Quasidiagonality passes to inductive limits, so long as
the connecting maps are injective.^3

Proof. Assume A = LJ An where An C An+l are QD subalgebras of A.
Apply Arveson's Extension Theorem to all c.c.p. maps An-+ Mk(n)(CC). 0

Here are two trivial permanence properties (proofs are left to the reader).

Proposition 7.1.10. Subalgebras of QD algebras are also QD.

Proposition 7.1.11. If each An is QD, then so is ITn An and {hence)
EBnAn.

The next permanence property is a little harder.^4

Proposition 7.1.12. The minimal tensor product of two QD C*-algebras
is again QD.

Proof. Let cpi: A -+ Mk(i) (CC) and 1/y B -+ Mz(j) (CC) be c.c.p. maps which
are asymptotically multiplicative and isometric. The obvious thing to do is
to consider

cpi 01/;j: A 0 B -+ Mk(i) (CC) 0 Mz(j) (CC)

and hope that we get an asymptotically isometric sequence of maps ( asymp-
totic multiplicativity is clear). This is essentially true but requires proof.

Let

<I>. A -+ IT Mk(i) (CC)

. EB Mk(i) (CC)

and
w· B-+ ITMz(j)(CC)
· EBMz(j)(C)

be the maps obtained by composing EB cpi: A -+ IT Mk(i) (CC) and, respec-
tively, EB 1/;j: B -+ IT Mz(j) (CC) with the quotient maps

IT

ITMk(i)(CC).. IT ITMz(j)(CC)

Mk( i) (CC) -+ EB Mk( i) (CC) and, respectively, Mz(j) ( C) -+ EB Mz(j) ( C)"

Note that and W are injective *-homomorphisms. Hence the tensor prod-
uct homomorphism

<I>®W:A®B-+ (ITMk(i)(CC)) 0 (ITMz(j)(CC))

EB Mk(i) (CC) EB Mz(j) (CC)

3The assumption of injective connecting maps is necessary -see Remark 17.3.3.