1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

20 1. Fundamental Facts

p1-= 1 - P. It follows from the identity

(P1-p(a)P)*(P1-p(a)P) = V(rp(a*a) - rp(a*)rp(a))V*

that

llP1-p(a)Pll = llr,o(a*a) - rp(a*)rp(a)ll^1 /^2 ,

for all a EA.

Now write .C = P .C EB p1-.C and decompose the representation p accord-

ingly. That is, consider the matrix decomposition

( ) _ [ p(a)11 p(a)i2 ]

p a - p(a)21 p(a)22 '

where p(a)21 = p1-p(a)P and p(a)i2 = p(a*)~h· Thanks to orthogonal

domains and ranges, the norm of the matrix

[

0 p(a)i2 ]

p(a)21 0

is equal to ~17.p(a).

Now comes the trick. We consider the Hilbert space p1-.C EB P £, and the

representation p': A---+ lBl(P1-.C EB P .C) given in matrix form as

p'(a) = [ p(a)22 p(a)21 ].

p(a)i2 p(a)11

Using the obvious identification of the Hilbert spaces

P .C EEl ( E9 p1-.C EEl P .C) and Elj .C = Elj(P .C EB p1-.C),

N N N

a standard calculation shows that

/lp(a)11 EEl p'^00 (a) - p^00 (a)/I :::; 'T/cp(a)

for all a EA, where p'^00 = EBw p' and p^00 = EBw p. Note also that p(a)11 =
Vrp(a)V.
Let C = rp(A) +lK(1i) and observe that C is actually a separable, unital
C-subalgebra of IBl(1i) with n(C) ~ A (again, n: IBl(1i) ---+ Q(1i) is the
quotient map). Note that /, EB p'^00 o 7r is approximately unitarily equivalent
relative to the compacts to 1,, where 1,: C '---+ IBl(1i) is the inclusion. Let
Wn: 1i---+ 1i EEl (EBw(P1-£ EEl PC)) be unitaries such that

llr,o(a) EEl p'^00 (a) - Wn<p(a)W~ll---+ 0

for all a EA.

Let

V: 1i EEl (Elj(P1-.C EEl P .C)) ---+ Elj .C

N N