12.2. The Haagerup property 359Observe that the semidirect product Z^2 ><l SL(2, Z) of Z^2 by SL(2, Z)
does not have the Haagerup property (by Theorem 12.1.10 and Proposi-
tion 12.2.3). Thus, an extension of groups with the Haagerup property need
not have the Haagerup property. The group Z^2 ><l SL(2, Z) also serves as
a counterexample to preservation of the Haagerup property under amalga-
mated free products; indeed,
Z^2 )<] SL(2, Z) ~ (Z^2 )<] (Z/4Z)) *z2~(Z/2Z) (Z^2 )<] (Z/6Z)).The Haagerup property for von Neumann algebras. Let M be a von
Neumann algebra with faithful normal tracial state Tande: M--+ M be a
T-preserving u.c.p. map. Then e extends to a contraction on L^2 (M, T).
Definition 12.2.14. Let M be a von Neumann algebra with a faithful
normal tracial state T.^8 We say that M has the Haagerup property if there
exists a net of T-preserving u.c.p. maps ei on M such that each ei extends
to a compact operator on L^2 (M, T) and ei --+ idM in the point-ultraweak
topology.
Theorem 12.2.15. A group r has the Haagerup property if and only if
L(r) has the Haagerup property.
Proof. Let r be a group with the Haagerup property and take a net !.pi
of positive definite functions as in Definition 12.2.1. Then the multipliers
ei = m'Pi (Theorem 2.5.11) are readily seen to satisfy the definition of the
Haagerup property for L(r).
To prove the converse, assume that L(r) has the Haagerup property and
take a net ei as in Definition 12.2.14. Let V: £^2 (r) --+ £^2 (r) ® £^2 (r) be the
isometry given by V 8t = 8t ® 8t and
er: L(r) --+ L(r) ® L(r) c JB( £^2 (r) ® £^2 (r))
be the normal *-homomorphism given by er(.A(s)) = .A(s) ® .A(s). It is easy
to check that
V*(.A(s) ® .A(t))V = { .A~s) ~!:; ~'.
We set !.pi(s) = T(.A(s)Bi(.A(s))) and observe that mcpi(a) = V(id ® ei) 0
er (a) V. It follows that each !.pi is positive definite,
limsup l1.pi(s)I :S limsup llBi(.A(s))ll2 = 0,
S->00 s->oo
and limi !.pi ( s) = 1 for every s E r. This proves the Haagerup property for
r. o
8 As with property (T), the definition of the Haagerup property does not depend on the choice
of the faithful normal tracial state -see [90] for more. Thus, for group von Neumann algebras we
can (and will) always take the canonical trace.