7.1. The defi.nition, easy examples and obstructions 241is also injective. Since injective representations of C* -algebras are always
isometric, it follows that the ( c.c.p.) tensor product map
(ffi Cfi) 0 (E91fij): A 0 B--+ ( IJ Mk(i)(C)) 0 ( IJMz(j)(CC))
is also isometric. Using the identification
( IJMk(i)(CC)) 0 ( IJMz(j)(CC)) ~ nMk(i)(CC) 0Mz(j)(CC),
i,Jone can find an asymptotically isometric sequence by taking direct sums of
maps of the form Cfi 01fij. D
Remark 7.1.13. It isn't known if the maximal tensor product of QD alge-
bras is again QD.
We will see more examples and permanence properties in the exercises
and later sections of this chapter. However, we now wish to discuss the
only two known obstructions to quasidiagonality (which will give us lots of
examples of nonquasidiagonal C*-algebras).
Obstruction 1: Every QD C*-algebra is stably finite.
Recall that an isometry s is called proper if 1 - ss* =/= 0.Definition 7.1.14. A unital C*-algebra A is stably finite if Mn(C) 0 A
contains no proper isometries, for every n E N. A nonunital algebra is
stably finite if its unitization is.
Proposition 7.1.15. Every QD C*-algebra is stably finite.
Proof. The proof boils down to the following routine exercise: If Tn E
Mk(n) (CC) and llT~Tn - lk(n) II --+ 0, then llTnT~ - lk(n) II --+ 0 too.
Hoping for a contradiction, we assume A is quasidiagonal and that it
contains a proper isometry s E A. (Since Mn(A) is QD whenever A is, we
may assume A contains the proper isometry.) Let Cfn: A--+ Mk(n)(CC) be
asymptotically multiplicative, asymptotically isometric u.c.p. maps. Then
1 = Cfn(ss) >=:::! Cfn(s)cpn(s) while Cfn(s)cpn(s) can't get close to 1 since
1 - ss =/= 0. This contradicts the exercise above. D
It follows that many standard examples of C* -algebras are not QD: the
Toeplitz algebra, IB(1i), Cuntz algebras, and many others (cf. Sections 4.5
and 4.6).
One may wonder whether or not there is a converse to the previous
proposition and the answer is: "No; yes; we don't know; maybe." More pre-
cisely, in general it is not true that every stably finite C* -algebra is QD -