1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.1. Kazhdan's property (T) 345

We identify the Pontryagin dual of Z with 'JI' = IR/Z by the pairing

Z x 'JI' 3 (n, t) 1-+ e^2 7rint (where Jr = 3.14 · · · in the exponent has noth-

ing to do with the representation Jr and where i = H). It follows that

C*(Z^2 ) ~ C('JI'^2 ) and the representation Irlz2 gives rise to a *-representation

a-: C('JI'^2 ) ---+ IIB(H) such that o-(zi) = Ir(hi), where Zi(t) = e^2 7riti for t =

(ti, t 2 ) E 'll'^2. We denote byμ the regular Borel probability measure on 'JI'^2

defined by

}y2 r Jdμ= \o-U)~,~).

Note that the formula makes sense for every bounded Borel function f on

'll'^2. Seto= (0, 0) E 'll'^2. Since o-(X{o}) is the orthogonal projection onto the

space of Z^2 -invariant vectors (which is zero by assumption), we have that

μ({o}) = llo-(X{o})~ll^2 = 0. Setting Bo:= {t: ~z1(t)::::; 0 or ~z2(t)::::; O}, we

claim that μ(Bo) ::::; 82. Indeed, for i = 1, 2, one has

82 2 II Ir( hi)~ - ~11^2 = j lzi(t) -11^2 dμ(t) 2 2μ( {t: ~zi(t) ::::; O} ).

The natural action of SL(2, Z) on Z^2 gives rise to an action of SL(2, Z)

on 'JI'^2 : For g E SL(2, Z) and f E C('ll'^2 ), one has

Ir(g)o-(j)Ir(g)-^1 = o-(g. f),

where §1 (ti, t2) = (ti, -ti+ t2) and §2(t1, t2) = (t1 - t2, t2)· We claim that

for i = 1, 2 and any Borel subset B C 'll'^2 , one has lμ(fliB) - μ(B)I < 28.

Indeed, for any f E C('JI'^2 ), one has

I j(?Ji · !-J)dμI = l\a-(J)Ir(gi^1 )~,1r(gi^1 )~) - (a-(J)~,~)I::::; 20\Jlloo·

It follows that \μ(§iB) - μ(B)I =If (§i · XB - XB)dμI ::::; 28.

Now consider the measurable partition

given by

4

'JI'^2 =[-1/2,1/2)^2 = {o} LJ lJ Bk

k=O

Bo= {(ti, t2) : ltil 2 1/4 or lt2\ 2 1/4},

Bl= {(ti, t2) : \t2\ ::::; lt1\ < 1/4 and tit2 > O},

B2 = {(t1, t2) : lt1\ < \t2\ < 1/4 and tlt2 2 O},

Bs ={(ti, t2) : \t1\ ::::; \t2\ < 1/4 and tit2 < O},

B4 ={(ti, t2) : \t2\ < ltil < 1/4 and tit2 ::::; O}.

It is easy to check that