On the Characters and the Plancherel Formula of Nilpotent Groups ...
On the Characters and the Plancherel Formula of Nilpotent Groups ...
On the Characters and the Plancherel Formula of Nilpotent Groups ...
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302 DUDLEY<br />
hence probability 1. Hence almost every w belongs to Cm(e) for some n.<br />
Then since L o P, is continuous, <strong>the</strong>re is an 01 > 0 such that if<br />
x, y E A <strong>and</strong> (1 x - y 11 < (II, <strong>the</strong>n<br />
I Q) -L(Y) I b) < I Wn(x - Y)) b) I + 2Q9aLC < E-<br />
Since E was an arbitrary positive number, (b) is proved.<br />
(b) z- (c): given E > 0 we use uniform continuity on C with<br />
probability 1 to infer that for some 6 > 0,<br />
Pr (sup {I L(x - y) I : x, y E A, /I x - y II < 63 > e) < E.<br />
We choose a finite-dimensional subspace F such that F n C is within<br />
6 <strong>of</strong> every point <strong>of</strong> C. Let P be <strong>the</strong> projection onto F. Then by Pro-<br />
position 4.1<br />
since, for any x in C, <strong>the</strong>re is a y in F n C with 11 x - y 11 < 8 <strong>and</strong><br />
Ply = 0. Thus<br />
Pr(j(L--LOP)-(C)I>,E) 0 <strong>the</strong>re<br />
is an n such that<br />
Pr (E(Q,“C) > c/2) < c/2.<br />
Now <strong>the</strong> operator norm 11 PmlQn I/ -+ 0 as VI -+ co since Qn has finitedimensional<br />
range <strong>and</strong> Pml-+ 0 pointwise. Hence E(P,-LQ,,C) -+ 0<br />
in probability as m -+ co. Also<br />
Pr (~(P,IQmLC) < 42) 2 Pr (yQfl’C) < e/2) >, 1 - c/2,<br />
so, for m large enough,<br />
<strong>and</strong> (d) holds.<br />
~(P,IC) < ~(P,IQnC) + ~(P,IQdC);<br />
Pr (L(P,T) > c) < E