1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.3. Weak amenability

be an isometry such that, for each s Er,

1

VFb"s = /IDT I: 88 -lp 0 b"p 0 b"cr(p)-lscr(s-lp)·

v IFI pEF

367

Setting 'lj;p(x) = V_F(le2(r/A) 0 x)VF for x E IIB(f^2 (F)) 0 C~(A), it is routine

to check that

1/Jp(ep,q 0 AA( a))= l~l 8acr(q)-1q,cr(p)-1pA(o-(p)aO"(q)-^1 ) E C~(r)

for every p, q E F and a E A. (As you probably guessed, 8acr(q)-1q,cr(p)-1P is
the Kronecker delta.) Hence, the range of 1/JF is contained in C~(r) and

JFnsFJ

1/JF o cpp(,.\(s)) = IFI ,.\(s)

for every s Er. Taking F to be a suitable F¢lner subset, we are done. D

Corollary 12.3.12. Let A be a co-amenable subgroup ofr. Then Acb(r) =
Acb(A).

Theorem 12.3.13. For C*-algebras Ai and A2 we have Acb(Ai 0 A2) =
Acb(Ai)Acb(A2). The same is true for von Neumann algebras (replacing 0
with®). In particular, Acb(ri x r2) = Acb(ri)Acb(r2), for discrete groups
ri and r2.

This theorem holds for arbitrary operator spaces (with the obvious def-
inition of Acb)· Hence the proof requires some operator space results, which
we now establish.

Let A and E be operator spaces with dim(E) < oo. Then there is a
natural inclusion A0E '---+ CB(A, E), where the duality between CB(A, E)
and A 0 E is given by (cp, u) = ~k fk(cp(ak)) for cp E CB(A, E) and u =
~k ak 0 fk E A 0 E. We denote by Ai§ E the space A 0 E equipped with
the norm induced from CB(A, E). An element f E Mn(E) is written as
f = [fi,j] E Mn(E), where (j,x) = ~fi,j(Xi,j) for x = [xi,j] E Mn(E). For
a= [ai,j] E Mn(A) and f = [fi,j] E Mn(E), we define a~f = ~ai,j®fi,j E
Ai§ E*.

Lemma 12.3.14. Let A and E be as above. Then, (Ai§ E) = CB(A, E)
canonically isometrically and the set

{a~f: n EN, a E Mn(A), f E Mn(E) with JJaJJMn(A) < 1, JJfJJMn(E) < l}

is dense in the unit ball of A i§ E*. In particular, the map

Ai i§ Ei x A2 i§ E2, 3 (ui, u2) f---t ui x u2 E (Ai 0 A2) i§ (Ei 0 E2)*,
defined by (ai 0 Ji) x (a2 0 h) = (ai 0 a2) 0 (ii 0 h) on elementary tensor
products, satisfies JJui x u2 JJ ~ JJui IJ JJu2 JJ.