348 12. Approximation Properties for Groupsfor any unitary representation (7r, J-t) of I' and e E H, where Pi is the
orthogonal projection onto the subspace of Hi-invariant vectors. Let l be
such that r = Hi(I) · · · Hi(Z)G and set c = K,/(7l + 2). We will prove that
(E, c) is a Kazhdan pair for r. Suppose by contradiction that there exists
a unitary representation ( 1f, H) of r having no nonzero invariant vector but
having an (E,c)-invariant vector fo E 1-i.
Let Pi be the projection onto the Hi-invariant vectors and set T = (Pi+
P2)/2. Observe that Pi and T commute with 7r(G). Since llfo-Pifoll::::; c/K,,
we have
1
(Tfo,fo) = 2(llP1foll^2 + llP2foi1^2 ) 2: 1-(c//'i,)^2
and so the spectrum of T intersects [1 - ( c / K, )^2 , 1 J. But on the other hand,
1 is not in the point-spectrum of T, since I' is generated by H1 and H2
and 1f does not have a nonzero invariant vector. It follows that there exists
0 < 8 < (c/K,)2 such that the spectral projection Q = X[I-(e/A;)2, 1 _ 0 1(T) is
nonzero.
We claim that llQ - 7r(h)Qll ::::; 2V'2c/K, for every h E H1 U H2. Let
( E QH be a unit vector. Then we have
1 - (c/ /'i,)^2 ::::; (re,() = llP1(ll^2 /2 + llP2e11^2 ;2 ::::; 1;2 + llPie11^2 /2,
and hence llPi(ll^2 2:1-2(c/K,)^2. Since7r(Hi) commutes with Pi, this implies
that
Vh E Hi, 11e - 7r(h)(ll = llP/-(-7r(h)P/-ell::::; 2./2c//'i,.
This proves the claim.
Note that 7r(h)Q - Q7r(h) = QJ7r(h)Q - Q7r(h)QJ and that QJ7r(h)Q
and Q7r(h)QJ have mutually orthogonal domains and ranges. It follows
from the inequality llQJ_7r(h)Qll ::::; llQ-7r(h)Qll that for all h E H1 U H2 we
have
Since r = Hi(I) · · · Hi(Z)G and 7r(G) commutes with Q, it follows that
sup llQ - 7r(s)Q7r(s)*ll::::; 2l./2c/K, =:co.
sErNow, let C = convw{ 7r(s)Q7r(s)* : s E I'} c llll(H); this set is ultraweakly
compact and r acts continuously on it by conjugation. By Zorn's Lemma,
there exists a subset Co of C which is minimal among the nonempty r-
invariant ultraweakly-closed convex subsets of C.^5 Since the r-action pre-
serves the norm, all elements of Co have the same norm, say c. Pick some
SE Co. We note that llQ - Sii ::::; co and for all h E H1 U H2,
(*) !IS - 7r(h)Sll::::; 2co + llQ - 7r(h)Qll::::; (4l + 2)./2c/K, =: c1.
(^5) !f dim 1i < oo, then Co is a singleton, and we are essentially done here.