1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

11.4. Excision and Papa's technique

unit of this matrix algebra, then cutting an element x E J, we have

qxq = ( f Ai) x ( f fj,j)

i=l J=l

m

= L Vivi XVjVj

i,j=l

m

~ L Vi(YiP)*x(yjp)vj

i,j=l

m

i,j=l

m

= L cp(yiXYj)fi,j·

i,j=l

331

In both of the approximations above, the norm difference is less than c; /2;
hence m

llqxq - L cp(yixyj)fi.jll < s.
i,j=l
The only thing left to notice is that the matrix of P7rcp(x)P with respect
to the orthonormal basis {Yi}i= 1 is just [(7rcp(x)yj,Yi)]i,j = [cp(yiXYj)]i,j·
Hence we can identify PJIB(1tcp)P with the matrix algebra C*({fi,j}) in such
a way that P7rcp(x)P 1--+ '2:::7,j= 1 cp(yiXYj)fi,j, for all x EA. This completes
the proof. 0

With a tiny approximation argument, the theorem above is easily seen
to hold for arbitrary finite-rank projections P.

Remark 11.4.5 (Commutator estimates). An extremely important aspect
of the result above is that the projections Pi(P) almost commute with ele-
ments in A that almost commute with P. More precisely, we have

\[u,pi(P)]\^2 :S: l[P,7rcp(u)]l\^2 + 2\IPi(P)upi(P)-Pi((u))I\

l\[u,pi(P)Jll~,cp :S: l\Pi(P)\l~,cp (l\[7rcp\f~\l{]I\§ +4llPi(P)upi(P)-Pi(<I>(u))I\),