1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

4.6. Cuntz-Pimsner algebras 149

In particular, the covariant representation (Q1o1r:F(7i)' S) of Jt on O(Jt) is

universal.

Proof. The proof is similar to that of Theorem 4.6.18, except J(7r,r) = 17-i

by Lemma 4.6.15 and covariance. Indeed, with the same notation, it suffices

to show that Q is injective on B-::;.n for every n. As before, this requires

induction.

First, Q is injective on B-:::,_o f::! A by assumption. Now, let n ;::::: 0 and

suppose that Q is injective on Bsn· Then, by Lemma 4.6.16, we have the

commutative diagram

o ,.. Bn+i ,.. Bsn+i ,.. B-::;.n/(Bsn n Bn+i) ,.. o

lQ lQ lQ

0 ----+ Bn+l ----+ B-::;.n+i ----+ B-:::,_n/ (B-::;_n n Bn+1) ----+ 0

whose rows are exact. Since Bn+l f::! JK(Jt®(n+l)) f::! Bn+l, the left vertical

arrow is injective. The right arrow is injective since Q is injective on B<n,

by the induction hypothesis, and Q maps B-::;.n n Bn+l f::! JK(Jt©n 17-i) o~to

B<n - n Bn+l· Hence, the middle arrow is also I injective by the 5 Lemma.

This completes the proof. D

We have several corollaries of the gauge-invariant uniqueness theorems.

Corollary 4.6.21. Let 1t be a C* -correspondence over A. Let B C A and

JC C 1t be a closed subspace such that BJCB C JC and (~, rJ) E B for every

~, fJ E JC. Viewing JC as a C* -correspondence over B, we have a natural

inclusion T(JC) C T(Jt).

Corollary 4.6.22. Let 1t be a C* -correspondence over A. Let a be a *-

automorphism of A and U be an isometric surjection on 1t such that

U(a~b) = a(a)U(~)a(b) and (U~, UrJ) =a((~, rJ))

for every~' rJ E 1t and a, b EA. Then, there exists a *~automorphism (called

a Bogoljubov automorphism) au on T(Jt) such that

au( a)= a(a) and au(Tt,) = Tu(E,)

for every a E A and~ E Ji. The same thing holds for O(Jt).

Proof. ;Let a and U be given a~ above. Then, the pair of maps a: A 3
a f--> a(a) E T(Jt) and To U: 1t 3 ~ f--> Tu(E,) E T(Jt) is a representation
of 1t and induces a *-homomorphism au: T(Jt) ---+ T(Jt). The same con-
struction for a-^1 and u-^1 gives rise to the inverse of au and hence au is an
automorphism. To prove the assertion for O(Jt), we have to check the co-
variance property of the representation (a, So U) of 1t into O(Jt). For this,