148 4. Constructions
Here is the gauge-invariant uniqueness theorem for Toeplitz-Pimsner al-
gebras.
Theorem 4.6.18. Let 1i be a C-correspondence over A and ('1r,T) be a
representation of 1i such that ?T is faithful. Assume that
?T(A) nspan{T(e)T(77): e,77 E Ji}= {0}
and ( ?T, T) admits a gauge action. Then, the representation ( ?T, T) is univer-
sal. In particular, the representation ( 1TF(1i), T) of 1i on T(H) is universal.
Proof. Let (7r,T) be a universal representation with gauge action ~- By
Proposition 4.5.1, it suffices to show that the canonical surjection
Q: C(7r, 7)-+ C(?T, T)
is injective on the fixed point algebra C*(7r, 7)/3. By Lemma 4.6.17, it suffices
to show that Q is injective (isometric) on B~n for every n. We prove this by
induction. First, Q is injective on B~o ~A by assumption. Now, let n 2:: 0
and suppose that Q is injective on B~n· Then, by Lemma 4.6.16 and the
assumption that I(7r,r) = {O}, we have the commutative diagram
0 ____,. Bn+ - 1 ____,. B -~ n+ 1 ____,. B -~ n ____,. 0
lQ lQ lQwhose rows are split exact. Since Bn+l ~ IK(Ji®(n+l)) ~ Bn+l, the left
vertical arrow is injective. The right arrow is injective by our inductive
hypothesis. Hence, by the 5 Lemma, the middle arrow is injective too. D
Definition 4.6.19. Let 1i be a C-correspondence over A. A representa-
tion (?T, T) of 1i is covariant if ?T(a) = cr 7 (a) for every a E I1i =An IK(H),
where err is the -homomorphism defined in Proposition 4.6.3. A covariant
representation ( 7r, 7) of 1i is universal if for any other covariant representa-
tion ( ?T, T) of Ji, there is a (continuous) -homomorphism from C ( 7r, 7) to
C*(?T, T) sending 7r(a) to ?T(a) and 7(e) to T(e).
We note that the representation (Q1o?TF(1i)' S) of1i on O(H) is covariant
and admits a gauge action. Universal covariant representations also admit
gauge actions.
Cuntz-Pimsner algebras also enjoy gauge-invariant uniqueness.Theorem 4.6.20. Let 1i be a C*-correspondence over A and (?T, T) be a
covariant representation of 1i such that ?T is faithful. Suppose that ( ?T, T) ad-
mits a gauge action. Then, the covariant representation ( ?T, T) is universal.