1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

152 4. Constructions

(See Exercise 4.6.4.) Hence 'H&;;B is a C*-correspondence over A&;;B. This

correspondence is called the exterior tensor product of 'H and B.

Lemma 4.6.24. Let H be a C-correspondence over A and B be a C-
algebra. There are natural isomorphisms T(H &;; B) ~ T(H) &;; B and O(H &;;
B) ~ O(H) &;; B.

Proof. If we denote by r: H <--+ T(H) the canonical inclusion, then r&;;idB
is a representation of the C -correspondence 'H &;; B into T(H) &;; B which
admits a gauge action /z&;;idB. Observe that A&;;B does not contain nonzero
"compact operators," because the conditional expectation E7-l&;;idB: T('H) &;;
B --t A &;; B kills all such elements. Hence the first assertion follows from
Theorem 4.6.18. The proof of O(H &;; B) ~ O(H) &;; B is similar. It suffices
to show that the pair A&;; B <--+ O(H) &;; B and S &;; idB: H &;; B <--+ O(H) &;; B
is covariant. Since e$.®ii,'17®b = ef.,'17 &;; ab, we have JK('H &;; B) = JK('H) &;; B
and
I1-l®B = (JK('H) &;; B) n (A&;; B) c IIB('H) &;; B.
Let x E I7-l@B be given and w be any state on B. We note that (id&;;w)(x) E
lK('H) n A= I1-l. Since o-s@idB = o-s &;; idB, we have

(id&;;w) oo-s@idB(x) = o-s((id&;;w)(x)) = (id&;;w)(x)

by covariance of S: H <--+ O(H). Since w was arbitrary, o-s@i<lB(x) = x. 0

We've finally come to the point of this section!

Theorem 4.6.25. Let'H be a C*-correspondence over A. Then, the Toeplitz-

Pimsner algebra T(H) is nuclear (resp. exact) if and only if A is.

Proof. The "only if' direction is trivial since there is a' conditional expec-
tation T(H) --t A. So, assume A is nuclear (resp. exact). Let n: T(H) --t D
be a -homomorphism; it turns out that 7r is nuclear if nlA is (and this im-
plies both the nuclear and exact cases). Let B be any C-algebra. Since the
*-homomorphism if= (nlA) &;; idB: A&;; B --t D ®max B is continuous, the
representation

T: 'H &;; B 3 L ~k &;; bk c-+ L n(r(~k)) &;; bk ED ®max B

is also continuous and well-defined. Hence, by universality of T(H &;; B), the
representation (if, 7) induces a *-homomorphism T(H &;; B) into D ®max B.
Combined with the previous lemma, this shows that n &;; id is continuous
from T(H) &;; B into D ®max B. Thus Corollary 3.8.8 yields the desired
conclusion. 0

It follows that O(H) is nuclear (resp. exact) whenever A is nuclear (resp.
exact), since nuclearity (and exactness) pass to quotients (Corollaries 9.4.3