1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

146 4. Constructions

Let ( 7r, T) be a representation of a C -correspondence 1{ over A. Then,
for every n 2: 1, we define Tn: 1-l®n-- C*(7r, T) by

Tn(6 0 · · · 0 ~n) = T(6) · · · T(~n)
on elementary tensors. It is not hard to see that Tn(μ)Tn(v) = 7r( \μ, v))
for everyμ, v E H®n and the pair (7r, Tn) is a well-defined representation of
H®n on C(7r,T). We define C-subalgebras Bn C C(7r,T) by Bo= 7r(A)
and
Bn = span{Tn(μ)Tn(v)*: μ, v E H®n} = O"n(OC(H®n))

for n 2: 1, where O"n is the *-homomorphism defined in Proposition 4.6.3.
Recall that O"n is injective if 7r is. Now define an ideal J(7r,r) of A by

1(7r,r) = 7r-^1 (7r(A) n B1) c A

and an increasing sequence of subspaces B<n of C* ( 7r, T) by

B~n =Bo+ B1 + · · · + Bn C C*(7r, T).

Lemma 4.6.15. Suppose that 7r is faithful. Then, 1(7r,r) c ht· Moreover,

we have 7r(a) = O"r(a) for every a E I(7r,r)·

Proof. Let a E I(7r,r) and choose x E OC(H) such that 7r(a) = O"r(x). Then,
for any ~ E 1{ we have

T(a~) = 7r(a)T(~) = O"r(x)T(~) = T(x~).
Since T is injective, a~ = x~ for all ~ E H. This implies a = x E An OC(H) =
ht· D
Lemma 4.6.16. For every n 2: 1, the subspace B~n is a C -subalgebra of
C(7r, T) and Bn is an ideal of B~n· Moreover,

B~n n Bn+l = Bn n Bn+l = O"n(OC(H®n I(7r,r))).

Proof. Recall the general fact that if C and J are C -subalgebras of D such
that CJC c J, then C + J is a C-subalgebra of D. (Indeed, it is clear
that C + J is a C -algebra containing J as an ideal. Since the image of
C in C + J / J is a dense C-subalgebra, it must coincide with C + J / J, or
equivalently C + J = C + J.) Hence, by induction, it suffices to show that
B~n-1BnB~n-1 C Bn for every n 2: 1, which is obvious from the definition.

Letμ, v E 1i®n and a E I(7r,r)· Then, O"n(eμa,v) E Bn, but also

O"n(eμa,v) = Tn(μ)7r(a)Tn(v)* E Bn+l

since 7r(a) E B1. This proves O"n(OC(H®n I(7r,r))) C BnnBn+l· Conversely, let
x E B~n n Bn+l be given andμ, v E 1i®n be arbitrary. Then, x E B~n im-
plies that Tn(μ)xTn(v) E 7r(A), while x E Bn+l implies that rn(μ)xTn(v) E