360 12. Approximation Properties for GroupsIn recent years Popa has exploited the Haagerup property in combination
with relative property (T) - pitting one against the other, in some sense
- with remarkable success. Here we content ourselves with one striking
example of his work.
It is a fact that L(Z^2 ) is a Cartan subalgebra (Definition F.15) of L('ll^2 >4
SL(2, Z)). Evidently the normalizer generates everything, so we only have
to see why L(Z^2 ) is maximal. But in general, if A c r, x E L(r) n L(A)'
and x = I:sEr x(s)>..(s) is the Fourier expansion of x, then x(asa-^1 ) = x(s)
for every s Er and a EA (since [x, >..(a)] = 0). As x E £^2 (r), we conclude
that
suppx = {s Er: the set {asa-^1 : a EA} is finite}
for x E L(r) n L(A)'. It is now easy to deduce that L(Z^2 ) is a maximal
abelian subalgebra of L(Z^2 >4 SL(2, Z)).
The inclusion L(Z^2 ) c L('ll^2 >4 SL(2, Z)) has relative property (T) by
Theorem 12.1.10, while SL(2, Z) has the Haagerup property - this tension
yields the following theorem of Popa.^9
Theorem 12.2.16. Let r = Z^2 >4 SL(2, Z) and, to simplify notation, denote
the inclusion L('ll^2 ) c L(r) by A C M. If B c M is another Cartan
subalgebra with relative property (T), then there exists a unitary element
u E M such that u* Bu = A.
Proof. We are going to use Theorem F.12 and Lemma F.17. Take a se-
quence 'Pn of positive definite functions on SL(2, Z) as in Definition 12.2.1.
It is not hard to check that the map en: qr] __, qr]' defined by
en(A.(as)) = 'Pn(s)>..(as),
extends to a trace-preserving u.c.p. map on L(r). Since en __, id in the
point-ultraweak topology and B c NI has relative property (T), there exists
no such that llb - en 0 (b)ll2 < llbll/3 for all b E B. Let T E JB(L^2 (M)) be
the positive contraction given by T(x) = en-- 0 (x) for x EM (or equivalently
TOas = 'Pn 0 (s)Oas on £^2 (f)) and d = X[l/3,lj(T). Note that T and d commute
with the right A-action and hence d E (M, A), where (M, A) is the basic
construction (see Appendix F). Moreover, we have
Tr(d) = l{s E SL(2,Z): 'Pn 0 (s) 2:: 1/3}1 < oo
for the canonical trace Tr on (M, A). For any unitary element w E B, we
have
Ill -wdwlll :'S 1/3 +II~ -T~ll = 1/3 + llw - en 0 (w*)ll2 :'S 2/3
(^9) Combined with work of Gaboriau on ergodic theory, this result gives the first example of a
type II1-factor whose fundamental group is trivial ([160]).