54 2. Nuclear and Exact C* -AlgebrasIt follows that for each fixed pair g, h E I',
converges to (xo 9 , oh) = ahg-1 as k ----+ oo.
(9) * (1): Let <pn: C{(I')----+ Mk(n)(C) and 'I/Jn: Mk(n)(C)----+ C{(I') be
u.c.p. maps converging to idc~(r) in the point-norm topology. By Arveson's
Extension Theorem we may assume that the 'Pn 's are actually defined on
all of JIB(.€^2 (I')). In other words, letting n ='I/Jn o 'Pn, we have u.c.p. maps
n: JIB(£^2 (r)) ----+ C~(r) such that n(x) ----+ x for all x E C~(r). Taking
a point-ultraweak limit point of {n} (see Theorem 1.3.7), we get a u.c.p.
map : JIB(.€^2 (I'))----+ L(I') which restricts to the identity on C{(I'). This is
all we need to show amenability of r.
Let T be the canonical vector trace on L(I') and consider the state
on JIB(.€^2 (r)). Restricting to £^00 (I') c JIB(.€^2 (r)), we get an invariant mean.
Indeed, for any TE JIB(.€^2 (r)) ands Er we have
where the first equality uses the fact that restricts to the identity on C{ (r)
(hence C~ (I') falls in the multiplicative domain of ) and the second uses the
fact that Tis a trace. Thus if TE .€^00 (I'), we have ry(s.T) = ry(>. 8 T>.;) = ry(T),
since left translation is spatially implemented.
(10) * (1): Semidiscreteness allows one to construct a u.c.p. map
: JIB(.€^2 (r)) ----+ L(I') which restricts to the identity on L(I') (i.e., semidis-
creteness implies injectivity - Exercise 2.3.15). This is more than enough to
imply amenability of r, as we saw above. D
Remark 2.6.9. This theorem not only shows that amenable groups give
rise to a very natural class of nuclear C -algebras, but it also gives our first
examples of nonnuclear C -algebras (since there are plenty of nonamenable
groups).
Remark 2.6.10. A similar theorem holds in the locally compact case, but
not everything generalizes. For example, nuclearity or semidiscreteness need
not imply amenability in general; Connes proved in [41] that if Go denotes
the connected component of G and if G/Go is amenable, then C{(G) is
nuclear and L(G) is semidiscrete. In particular, all connected Lie groups
have nuclear reduced C*-algebras (though they need not be amenable).