4.6. Cuntz-Pimsner algebras 139semi-inner product defined by
(e &>r/,e0rJ) = (rJ',7rd(e,e))rJ).
We denote by 1i®BK the Hilbert C-module obtained from 1t8K by separa-
tion and completion. The vector in 1t ® B K corresponding to e ® rJ E 1t 8 K
is still denoted bye® rJ· Note that the kernel of 1t 8 K-+ 1t ®BK is
span{(eb) ® rJ - e ® (7rJG(b)rJ): e E 1t, rJ EK and b EB}.
There is a natural faithful -representation IIB(H) -+ IIB(H ®BK) given by
x 1-+ x ® 1, where (x ® l)(e ® rJ) = (xe) ® rJ for e E Ji, rJ E K and x E
IIB(H). In particular, 1t ®BK is an A-C C-correspondence. There is also
a natural *-representation 7r/G(B)' n IIB(K) -+ IIB(H ®BK), y 1-+ 1 ® y, where
( 1 ® y) ( e ® rJ) = e ® (YrJ). Indeed, we only need to check that 1 ® y is bounded:
n
[[(l®y) Lei®'r/i[[ = [[(Yf7,XYf7)[[^112 ,
i=l
where fl = [771 · · ·77nJT E JC,ffin, X = [7rK((ei,ej))]i,j E IIB(Kffin) and Y =
diag(y, · · · , y) E IIB(Kffin). But since X ;:=:: 0 commutes with Y, we have
n
[[(Yf7,XYf7)11 ~ [[Y[[^2 [[X^112 i7l\^2 = llYll^2 11Lei®77i[[^2.
i=l
The interior tensor product ®B has several nice properties, including
associativity and a distribution law with respect to direct sums. For each
e E 1t, we define a map Te: K -+ 1t ®E K by Te(77) = e ® rJ· Then,
Te E IIB(K, 1t ®BK) with T!(( ® 77) = 7rJG( (e, () )77· In particular,
T!Te = 7rJG( (e, e')) and TgtT{ = Be',e ® 1.
We note that for the identity correspondence A over A, there are natural
identifications
1t ®A A ~ 1t and A ®A 1t ~ 7r'J-l(A)1t.Lemma 4.6.1. Let 1t be a Hilbert A-module and let 6, ... , en, 771, ... , 77n E
1t. Then we have
n
II L eei,'f/i [\JK(?-l) = II [(ei, ej)]ij
2
[(77i, 77j)lif2
11Mn(A)'
i=l
where [ (ei, ej) ]ij and [ (77i, 'r/j) ]ij are positive elements in Mn(A).Proof. For every e E 1t, we denote by Te E IIB(A, 1t) the operator given by
Te(&) =ea. Then T!T'f/ = (e, 77) and Be,'f/ = TeT;. It follows that for every
e1, ... ,en,771, ... ,rJn E Ji,
n n
L:eei,'f/i = L:TeiT;i = T~T;,
i=l i=l