15.3. Examples 417This yields the second assertion. For the third assertion, we observe that
((ytx) (p )-((yt) (p) is nonzero only if p E t supp(x) and that for q E supp(x ),
one has
((ytx)(tq) = { min{ltqlA, lqJA}
0+ Jy(tq)x(q)[r. if y(tq) # x(q)-i,
if y(tq) = x(q)-i,
((yt)(tq) = { min{[tqJA, [q
0[A} + [y(tq)[r if tq E supp(y),
if tq fj. supp(y).
Hence for q E supp(x), one has
[((ytx)(tq) - ((yt)(tq)J :S JqJA + Jx(q)Jr = ((x)(q)
and the third assertion follows. Finally, since ((yts)(p) and ((yt)(p) are
nonzero only if p E supp(y) and
[((yts)(p) - ((yt)(p)[ =I min{[PJA, Js-iriPIA} - min{JP[A, JripJA}[ :S JsJ,
for p E supp(y), the fourth assertion follows. DProof of Proposition 15.3.6. With Lemmas 15.3. 7 and 15.3.8 in hand,
it is easy to verify the condition of Lemma 15.2.6. Indeed, one just has to
check the condition separately for x E YA and s E A, acting from the left or
~~M. D
Corollary 15.3.9. Let r = Y 2 A be the wreath product. Suppose that Y is
amenable and A is bi-exact relative to {1} (e.g., if A is hyperbolic). Then,
r is bi-exact relative to {1}.
Proof. Combine Lemma 15.3.5, Proposition 15.3.6 and Proposition 15.2.7.
D
Theorem 15.3.10. Let r = Y 2 A be the wreath product of an amenable
group Y by an exact group A. If N C pL(f)p is a von Neumann subalgebra
with a noninjective relative commutant, then N embeds in L(A) inside L(I').Proof. Combine Theorem 15.1.5 and Proposition 15.3.6. D
Corollary 15.3.11. Let r = Y 2 A be the wreath product of an amenable
group Y by an exact group A. If N c L(f) is a noninjective nonprime factor
whose relative commutant N' n L(f) is a factor, then there exists a unitary
element u E L(r) such that uNu* c L(A).Proof. Write N as a tensor product N = Ni ® N2 of type IIi -factors Ni
and N 2. Since N is noninjective, we may assume that N2 is noninjective.
By Theorem 15.3.10, Ni embeds in L(A) inside L(f). By Lemma F.18 and
Theorem F.20, we can find a unitary element u E L(f) such that uNiu* C
L(A). This implies uN2u* c (uNiu*)' n L(f) c L(A), by Theorem F.20.
Therefore, uNu* C L(A). D