310 10. Summary(1) (Connes's embedding problem) Every separable II1-factor embeds
into the ultraproduct Rw of the hyperfinite II1-factor.
(2) Every separable C*-algebra with the LLP also has the WEP (cf.
Chapter 13).
(3) C*(JF2 x lF2) = C*(IB'2) 0max C*(F2) is RFD.
( 4) Every tracial state on C* (F 00 ) is amenable.
(5) C*(lF2) @max C*(JF2) = C*(JF2) 0 C*(IB'2).
(6) C*(JF2) @max C*(JF2) has a faithful tracial state.
(7) Several others -see Section 13.3.Group theory experts seem to agree: Gromov has shown that nonexact
discrete groups exist (cf. [71]). We're willing to take their word for it, but
we would love to understand it.
Problem 10.4.5. Is there a simple proof of the existence of nonexact
groups? Is there an explicit example?
The fact that all linear groups are exact ([73]) suggests the following:
Problem 10.4.6. Is every residually finite group exact? Perhaps just
coarsely embeddable into a Hilbert space?
An affirmative answer to the next question would imply the existence of
hyperbolic groups which are not residually finite.
Problem 10.4. 7. Does the factorization property pass to arbitrary induc-
tive limits (i.e., allowing noninjective connecting maps) of discrete groups?On the other hand, every hyperbolic group is residually finite if our next
question has an affirmative answer.
Problem 10.4.8. Do hyperbolic groups have the factorization property?More generally, it would be very nice to find geometric or dynamical
proofs of the factorization property for particular examples.
Our final problems are for C* -fanatics; we aren't aware of any important
applications, but they are basic questions which should be resolved.Problem 10.4.9. Is the hyperfinite II1-factor QD?^3
Problem 10.4.10. Are separable exact QD C* -algebras AF embeddable?
Problem 10.4.11. Does local reflexivity imply exactness?
Kirchberg has suggested an affirmative answer to this last question.(^3) This seems unlikely, but a proof would be nice. On the other hand, if it is QD, then
every simple unital stably finite nuclear 0*-algebra would be QD (compare with Blackadar and
Kirchberg's questions at the end of Section 8.5).