- References 209
a E Mn(<C). For simplicity, let az = 'lj;(>..(Jz)). Then every az is a contraction
such that
((az@ 1@ l)17'P -17'P>.(Jz), (az Q9 1@ l)17'P -17'P>.Uz))
= <.p(aiaz) + >-(Jz)*<.p(l)>.(Jz) - <.p(az)* >-(Jz) - >-(Jz)*<.p(az)
:S ( <.p^0 'l/J) ( >. (!{ fz)) + X (f{ fz) - ( <.p^0 'l/J) ( >. Uz)) * >. Uz)
-(Jz)*(<.p^0 'lf;)(>-(fz))
:::; 3c:.
This implies that ( az@ 1@ 1 )77'P ~ 17'P>.Uz) for every l. We approximate 17'P by
17~ E .e; Q9 .e; Q9 Cc( G) of norm one such that ll77'P -17~ II < c: and view 17~ as an
elementinCc(G,£;@.e;). Now we define( E Cc(G) by((,8) = ll77~(,6)llc~®£~·
Thus, if 17~ = ~j,k~jl8l~k@(k,j, then ((,6)^2 = ~j,k l(k,j(,6)1^2. It follows that
((, ()L2(G)(x) = L ((,8)^2 = L((k,j * (k,j)(x) = (77~, 17~/(x) :::; 1
(3EGx j,kfor every x E Q(O). Fix ry E Vi and let x = s(ry). Then,
1 = (!{ * fz)(x) ~ ((az@ 1Q91)17~, 17~>.(Jz))(x)
since llflloo :S 11>-(J)ll for all f E Cc(G). For ,6 E Gs('Y)' we have
(17~>.(Jz))(,6) = L 17~(,ea-^1 )fz(a) = 17~(/3ry-^1 ) E .e; @£;.
aEGs('l')Therefore,
1~l((az@1 @1)17~,17~>.(Jz))(x)I
=I L ((az@ 1)17~(,6), (17~>.(fz))(/3))c~®£~ISince IKllL2(G) :S 1, this implies that (( ()(ry-^1 ) ~ 1 for all ry EK. D
- References
Theorem 5.1.6 can be found in [5], [74] and [130]. Proposition 5.2.1 comes
from [136]. Exactness of amalgamated free products follows from Dykema's
C*-theorem (Corollary 4.8.3); however, geometric proofs - using Yu's prop-
erty A -can be found in [16] and [187]. The compactification of a tree de-
scribed here is a special case of the Bowditch compactification (unpublished).
Theorem 5.4.1 follows easily from general results on Morita equivalence; see