4.6. Cuntz-Pimsner algebras 143Since F(1i) =span T(1i)1i^1590 , it coincides with the GNS Hilbert A-module
for (T(1i), EH) by uniqueness of GNS representation. This shows nonde-
generacy of EH. For assertion (3), observe that each "Yz is implemented by
a unitary operator Uz E JIB(F(1i)) defined by Uz = EBn>O zn on F(1i) =
EBn>O 1i®n. (Although z 1-Y Uz is not norm-continuous, Z-1-Y "'fz = AdUz is
continuous.) D
Since the Toeplitz-Pimsner algebra T(1i) is too large for many purposes,
we define the Cuntz-Pimsner algebra 0(1i) to be a natural quotient of T(1i).
Let IH = An JK(1i) in JIB(1i). Since IH is an ideal in A and F(1i)IH is a
JIB(F(1i)) invariant C-sub-correspondence of F(1i), the C-algebra
OC(F(1i)IH) = span{B(, 77 : ~' 'T/ E F(1i)JH}is an ideal of JIB(F(1i)).
Lemma 4.6.7. We have an inclusion OC(F(1i)IH) C T(1i). Moreover,
OC(F(1i)IH) is globally invariant under the gauge action.
Proof. Letμ E 1i®n, v E 1i®m (m, n 2:: 0) and x E IH. Then, we have
Bμx,v = TμPoxPoT: = Tμ(x - O"_r(x))T;.Since O"_r(OC(1i)) = span{T(T; : ~' 17 E 1i} C T(1i), we have Bμx,v E T(1i).
Since F(1i)h-r. is invariant under the unitary operators Uz defined in the
proof of Theorem 4.6.6, the second assertion follows. D
Denote by Qr: JIB(F(1i)) -t JIB(F(1i)) /OC(F(1i)lri) the quotient map and
write S( = Qr(T()· Since OC(F(1i)JH) is an ideal of JIB(F(1i)), it is an ideal
of T(1i) as well. Note that Qr is injective on A since A acts diagonally on
F(Ji) = EB 1i®n ·
Definition 4.6.8. Let 1i be a C-correspondence over A. The (augmented)
Cuntz-Pimsner algebra 0(1i) of 1i is Qr(T(1i)), the C-algebra generated
by A and {S(: ~ E 1i}.
We record the main identities that hold in 0(1i).Theorem 4.6.9. Let 0(1i) = C(A U {S( : ~ E 1i}) be the Cuntz-Pimsner
algebra of a C -correspondence 1i over A.
(1) For every a E <C, ~' 17 E 1i and a, b EA, we have
SaH 71 = aS( + S 71 , Sa(b = aS(b and Sf,S 71 = (~, 17).
(2) If a E IH, then we have
a= O"s(a),
where O"S: OC(1i) -t 0(1i) is the *-homomorphism given in Propo-
sition 4.6.3.