1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

138 4. Constructions

and note that Hean is canonically isomorphic to .e; Q9 1-i equipped with the

A-valued inner product

where the inner product on .e; is linear in the second variable. The same
thing holds if we replace .e; with £^2.

Let 1 E D C A be unital C -algebras with a conditional expectation E
from A onto D. Then, A is naturally a right D-module and (a, b) = E(ab)
is a D-valued semi-inner product. We denote by L^2 (A, E) the Hilbert D-
module obtained from A by separation and completion and by a E L^2 (A, E)
the vector corresponding to a E A. Left multiplication by elements in A
defines a *-representation 7fE: A-+ JBl(L^2 (A,E)); 1fE(a)b = :J. For eE =
i E L^2 (A, E), we have
E(a) = \eE, 1fE(a)eE)

for every a E A. Hence it is quite natural to call (7rE, L^2 (A, E), eE) the
GNS representation for (A, E). We say the conditional expectation E is
nondegenerate if 7f E is faithful (or, equivalently, a = 0 if and only if E ( xay) =
0 for all x, y E A). Since the conditional expectation E: A -+ D extends to
an orthogonal projection E from L^2 (A, E) onto eED, we have L^2 (A, E) ~
eED EB L^2 (A, E)^0 , where L^2 (A, E)^0 = ker E c L^2 (A, E). We note that
Anker Eis dense in L^2 (A, E)^0 • Indeed, if Xn EA and:£;;;-+ 'r/ E L^2 (A, E)^0 ,
then (xn -E(xn)Y'-+ 'r/· Since L^2 (A, E)^0 is invariant under 1fE(D), we may
restrict 1fEID to get a -representation 1f°EJ: D -+ lBl(L^2 (A, E)^0 ). In other
words, L^2 (A, E)^0 is a C-correspondence over D.
By definition, an A-B C -correspondence is a Hilbert B-module 1-i to-
gether with a faithful -representation 7fH : A -+ lBl ( 1-i); it is called a C -
correspondence over A when B = A. (NB: Sometimes these are called
C -bimodules in the literature. Also, some authors do not require faithful-
ness in the definition.) The -representation 7f7-1, is referred to as the left
action of A on 1-i. We often omit 1fH when there is no confusion. We say
that the C -correspondence 1-i is full if the ideal { ( e, 'r/) : e, 'rJ E 1-i} is dense
in B. It is nondegenerate if 1fH(A)1-i is dense in 1-i (in fact it equals 1-i,
by Cohen's factorization theorem - Theorem 4.6.4). If {1-ii}iEJ is a collec-
tion of A-B C -correspondences, then the direct sum E9 Hi is naturally an
A-B C -correspondence with left action given by 7f Ee Hi = E9 7fHi. The sim-
plest C* -correspondence is the identity correspondence which is the Hilbert
A-module A with the left action given by multiplication from the left.

Now we introduce the interior tensor product of C -correspondences.
See Chapter 4 of [114] for proofs of the following claims. Let 1-i, JC be,
respectively, A-Band B-C C-correspondences. Then, the algebraic tensor
product 1-i 8 K is naturally a right C-module, together with the C-valued