1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

298 9. Local Reflexivity

Corollary 9.4.5. A separable C* -algebra is type I if and only if every sub-

algebra is nuclear.

Proof. Recall Glimm's Theorem: A separable C* -algebra A is not type I

if and only if every UHF algebra arises as a subquotient of A (see [142,

Section 6.8]).

This implies, first of all, that subalgebras of type I are again type I.

Indeed, if a subalgebra were not type I, then it would have UHF subquotients

and hence the larger algebra would too. Since type I C* -algebras are nuclear

(Proposition 2. 7.4), this evidently implies the "only if" direction.

For the opposite direction, we assume A is not type I and show that it has

a nonnuclear subalgebra. By Glimm's Theorem, we can find a subalgebra

B C A which has a prescribed UHF algebra as a quotient. By Corollary

8.2.5, B must have a subquotient isomorphic to C{(lF2) (or any other exact

nonnuclear C*-algebra). Since nuclearity passes to quotients, this implies

B, hence A, contains a nonnuclear subalgebra. D

Finally, we present a folklore result (due to Tomiyama and Kirchberg)
on the ideal structure of minimal tensor products (cf. [186]). It is a non-
commutative analogue of the fact (definition) that rectangular open subsets
form a topological basis in Cartesian products.

Corollary 9.4.6. Let I <I (A® B) be an ideal. If A is exact, then

I = span{IA 8 IB : IA, IB are ideals such that IA 8 IB C I}

= span{ x ® y : x ® y E I}.

Proof. First observe that the closed linear spans on the right hand side are
equal and define a closed two-sided ideal, denoted by J, contained in I. We
will show that if z E (A® B) \ J, then z ~ I. (You may want to review
Exercise 3.4.2 before proceeding.)

Let O": A® B -+ JIB(H) be an irreducible representation such that J c
kerO" and O"(z) I-0. Letting O"A and O"B be the restrictions (Theorem 3.2.6),
it is clear that ker O" A® B +A® ker O"B C ker O". Since A is exact, A® B /A®
kerO"B = A® O"B(B). Since A is locally reflexive, A® O"B(B)/kerO"A ®
O"B(B) = O"A(A) ® O"B(B) (Corollary 9.1.5). It follows that

(A® B)/(kerO"A ® B +A® kerO"B) ~ O"A(A) ® O"B(B),

and hence O" factors through O"A(A) ® O"B(B). In fact, O" drops to an isomor-
phism on O"A(A) ® O"B(B). Indeed, since O"A(A)" and O"B(B)" are factors, we
have O"(A8B) ~ O"A(A) 80"B(B) and thus O"(A@B) ~ O"A(A) @O"B(B) (cf.
Exercise 3.4.1).