1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

416 15. Group von Neumann Algebras

Lemma 15.3.7. For g ={A}, one has

lim I supp(y)I = 0

yt->oo/Q ll((yt)ll.

Proof. We first claim that liillyt->oo/Q ll((yt) II = oo. Let R > 0 be given and

suppose yt E r is such that ll((yt) II :::; R. Then supp(y) C BR(A) U tBR(A)

and y(p) E BR(Y) for every p EA. Define y' E YA by y'(p) = y(p) for p E

BR(A) and y'(p) = e for p tJ. BR(A). Then, y = y'y" with supp(y') C BR(A)

and supp(y") c tBR(A). Hence, for the finite subset

E = {z E YA: supp(z) C BR(A) and z(p) E BR(Y) for every p EA}

of YA, we have

yt = y I frxt-1 ( y ") E u z z 'A II.

z',z"EE

This means that the subset nR = {yt Er: ll((yt)ll:::; R} is small relative

to g and the claim follows.

Let C > 0 be given and suppose yt Er is such that JJ((yt)ll :::; Cl supp(y)I.

Since ((yt)(p) 2:: 2C for p E supp(y) \ (B2c(A) UtB2o(A)), we have

I supp(y) \ (B2c(A) UtB2o(A))I:::; I supp(y)l/2.

This implies I supp(y)I :::; 4IB2o(A)I and yt E nR for R = 4CIB2o(A)I. By

the first part of the proof, {yt E r : I supp(y)l/ll((yt)ll 2:: c-^1 } is small

relative to g. D

Lemma 15.3.8. The following hold:

(1) ll((xyt) - ((yt)ll:::; ll((x)ll for every x, y E YA and t EA;

(2) ll((syt) - s.((yt)ll:::; lslAI supp(y)I for every y E YA ands, t EA;

(3) ll((ytx) - ((yt) II :::; ll((x) II for every x, y E YA and t E A;

(4) ll((yts) - ((yt)ll:::; lslAI supp(y)I for every y E YA ands, t EA.

Proof. Note that ((xyt)(p) - ((yt)(p) is nonzero only if p E supp(x). Also,

IC(xyt) (p)-((yt)(p) I

_ { I lx(p)y(p)lr - IY(P)lrl if p E supp(y) n supp(xy),

• ( ( xt) (p) otherwise
  :::; ((x)(p)

for p E supp(x). This yields the first assertion. For the second, observe that
((syt)(p) and (s.((yt))(p) are nonzero only if p E ssupp(y) and that

I(( syt)(p )-( s.((yt)) (p) I

=I min{IPIA, l(st)-^1 PIA} - min{ls-^1 PIA, l(st)-^1 PIA}I

::::: Isl-