1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

250 7. Quasidiagonal C* -Algebras

and hence V is a partial isometry. Thus VV is a finite-rank projection in
JE(ffi~ JC). A slightly unpleasarrt calculation, using the fact that Fk+lFk =
Fkl shows that the matrix of VV is given by

Go a1/2a1/2 0 1 o o

01 1;2 0 1;2 0 01 01 1120 1;2 2 0

o a;^12 ai^12 G2 o

0 0 0

We let P = VV* E JE(EB~ JC) and define a representation of Bon EB~ JC
by

7r(b) = O"o(b) EB <71-(b) n EB O"~(b) n EB··· EB <71(b).

Since Fo 2: Q, it is not hard to see that

JJP7r(b)PJJ 2: J[b[[ - c,

for all b E i, and hence we only have to show that P almost commutes with
7r(i).

For this it is helpful to look at the matrix description of P given above
and to write P = LD + D + U D where D is the diagonal of the matrix,
while LD and U D denote the lower and upper diagonals, respectively. Ev-
idently, it suffices to show that the three commutators [LD,7r(b)], [D,7r(b)]
and [UD, 7r(b)] all have small norms and this will follow from the fact that
we arranged

[[[Fj,O"i(b)][[::; o
n
and

l/O"i(b) - O"j+l (b)fl::; o,
n n
for all 0 ::; j ::; n and b E i, at the very beginning. Indeed, it is not hard to
see that

[[[D,7r(b)]J[::; 4o.

The harder estimates are the off-diagonal parts and here is where we use the
fact that o has not yet been prescribed.

Since c is fixed, we can, by Lemma 7.3.2, find a o such that [[ [X^112 , b] II ::;
c whenever [[[X, b][[ ::; o, 0 ::; X ::; 1 and [[bf[ ::; 1. Hence, taking o small
enough, we can ensure that

[J[<72.(b),Gin^12 Jll::; c

whenever [j - k I ::; 1. Using this, it is not too hard to show that the norms
of [LD,7r(b)] and [UD,7r(b)] are also small for all b E i. Indeed, if you write