1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

254 7. Quasidiagonal C* -Algebras

Remark 7.4.3. Free groups actually enjoy a stronger property: Every rep-

resentation can be approximated in the Fell topology ([15]) by finite rep-

resentations (i.e., representations factoring through finite quotients of lFn);

see [123].

Though it seems uninteresting at first, a major open problem is whether

or not Choi's result can be extended to lFn x lFn (since it's equivalent to

Connes's embedding problem).

Proposition 7.4.4. The following statements are equivalent:

(1) every finite van Neumann algebra with separable predual can be

embedded into the ultraproduct of the hyper.finite II1 -factor (this is

Connes's embedding problem);

(2) there is a unique C*-norm on C*(lFn) 0 C*(IFn);

(3) C*(lFn x lFn) is residually finite-dimensional.

Proof. For the equivalence of (1) and (2) see Theorem 13.3.1. Since they

both satisfy the same universal property, it is easy to see that

C*(lFn X lFn) ~ C*(lFn) @max C*(lFn)·

Thus (2) ==?- (3) is easy (since the minimal tensor product of RFD alge-

bras is easily seen to be RFD). So, assume C*(JFn x lFn) is residually finite-

dimensional and let 7r i : C* (JF n) @max C* (JF n) -----+ Mk( i) ( C) be a separating

family of representations. By finite-dimensionality, each of the 7ri 's must

factor through the minimal tensor product, and hence the quotient map-

ping C*(lFn) @max C*(IFn)-----+ C*(lFn) 0 C*(lFn) must be injective. D

Though residual finite-dimensionality remains open, it is not too hard

to show quasidiagonality of C*(IFn x lFn)·

Proposition '7.4.5. The full group C*-algebra C*(lFn x lFn) is QD.

Proof. Let C(lFn x lFn) C JB(H) be a faithful representation. We only
consider the case n = 2; the general case follows since C(JF 00 x JF 00 ) c
C* (lF2 x lF2).
Let U1, Vi E JB(H) be the image of the unitaries which generate the
"left" copy of lF2 and let U2, V2 E JB(H) be the image of the unitaries which
generate the "right" copy of lF2.
Since the unitary group of JB(H) is connected (Borel functional calculus),
we can find a norm continuous path from any unitary to the identity. That is,
we can find norm continuous maps Ui: [O, 1]-----+ U(H) and Vi : [O, 1]-----+ U(H),
where U(H) is the unitary group of JB(H) and i = 1, 2, such that

Ui(O) = 1, Ui(l) = ui, Vi(O) = 1 and vi(l) =Vi.