10.4: Open problems 309Tensor products. It isn't known if the maximal tensor product of QD
C* -algebras is again QD. For spatial tensor products, everything is fine (see
Proposition 7.1.12 and Exercise 7.2.6).Proposition 10.3.7. If A and B are QD, then so is A 0 B.Crossed products. Quasidiagonality of crossed products is poorly under-
stood. The only cases where we have complete information is A ><la Z where
A is either abelian or AF. (See Section 8.5 for precise statements.) Recall
that Rosenberg's conjecture asks whether C~(r) =CC ><Irr is always QD for
a discrete amenable group r. There are lots of open questions in this area.Free products. The reduced free product of QD C* -algebras is almost
never QD (C~(IF2) = C('JI') *r C('JI') is not QD, and many reduced free prod-
ucts contain this algebra as a subalgebra). A nice theorem of Florin Boca,
however, reveals a different situation for full free products ([23]).Theorem 10.3.8. If A and B are unital QD C* -algebras, then the full free
product A* B (with amalgamation over CC) is also QD.It isn't known what happens with more general amalgamation.
10.4. Open problemsA:q. affirmative answer to our first question would imply that every exact C* -
algebra with the local lifting property is nuclear (it is known that A ®rnax
Q(H) = A 0 Q(1i) {::} A is both exact and has the local lifting property -
see Section 13.1).
Problem 10.4.1. Let Q(1i) = JIB(1i)/JK(1i) be the Calkin algebra. Is it true
that A is nuclear if and only if A ®rnax Q(1i) =A 0 Q(1i)? · · ·
The following problem appears tantalizing at firs.t glance, but ·it's prob-
ably impossible. 'Problem 10.4.2. Is there :;i, simpler proof of the fact that the double dual
of a nuclear C*-algebra is semidiscrete?
Problem 10.4.3. Let N C M c JIB(H) be von Neumann algebras and
assume cp: JIB(H) -+ M is a u.c.p. map such that cplN = idN. Is the inclusion
map N '-+ M weakly nuclear? ·Perhaps the most important problem in this section is
Problem 10.4.4. The following statements are equivalent ([102]). Are
they true?