1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

158 4. Constructions

Let p = 1-T^2 (T)^2 E C(Ai E9A 2 , T) be the orthogonal projection onto

EB( 1-{i E9 (1-ii @n 1-ij) E9 ((1-ii @v 1-ij e ein 0 ejD) @v Ki))'

ii=j

and recall that 1-ii = eiD E91-if,. Let u = p(T + T*)p E C*(Ai E9 A2, T) and

observe that u is a self-adjoint partial isometry with u^2 = p that interchanges

pKi and p!C2:

where the arrows are either T or T*. We define u.c.p. maps

7/Ji: Ai 3 a~ pap+ uau E pC*(Ai E9 A2, T)p

and note that 7/Ji ( d) = 7/J2 ( d) for d E D.

Theorem 4.8 .. 2. There exists a u.c.p. map

'11: Ai *D A2--+ pC*(Ai E9 A2, T)p

such that '11 ( d) = 7/Ji ( d) for d E D and

'11( ai ···an) = 7/Ji 1 ( ai) · · · 7/Jin (an)

for ak E Af,k with ii -=I-· · · -=I-in. Moreover, there exists a *-homomorphism

7r: C*(w(Ai *D A2))--+ Ai *D A2

such that 7r o '11 =id. In particular, '11 is a complete order isomorphism.

Proof. We decompose pKi = Ki,i E9 Ki,2 E9 Ki,3 E9 Ki,4 as follows. The first

component is

Ki,i = 1-ii E9EB1-ii ®v 1-ij 0n · · · 0n 1-ij 0n 1-ii;

in other words, all the tensors beginning and ending with vectors from 1-ii

and having vectors from the 1-ik,'s in the middle. The second is

Ki,2 = (1-ii 0v 1-ij) E9 EB Hi ®v 1-ij 0n · · · ®v 1-if, 0n 1-ij;

this is very similar to Ki,i, except we begin with Hi and end with 1-ij. Recall
that 1-{i = ein E9 1-if' and note that no tensors in the first two components
contain eiD or ejD in the middle -only on the ends. Our next two compo-
nents take care of this. The third component is