- Cuntz-Pimsner algebras 153
and 9.4.4). Unfortunately there is presently no direct proof of this fact,
however.
There is another proof of nuclearity/exactness of T('H), which we now
sketch. By Theorem 4.5.2, it suffices to show that the fixed point algebra
T('H)'Y of the gauge action 1 is nuclear/exact. As in the proof of Theo-
rem 4.6.18, this reduces to showing that every B-:::,n is nuclear/exact. Since
each B-:::,n is a split extension of B-:::,n-1 by Bn s:! JK('H®n), we are further
reduced to the nuclearity /exactness of JK('H®n). The exercises below finish
off the proof.
Exercises
Exercise 4.6.1. A C-subalgebra Ao c A is said to be full if Ao generates
A as an ideal - i.e., span(AAoA) = A. Let Ao be a full and hereditary
C -subalgebra of A and I be an ideal of A. Prove that Ao n I is a full,
hereditary C-subalgebra of I. (Generous hint: Suppose, by contradiction,
that Ao n I is not full in I. Then, there is a nonzero nondegenerate -
representation 7r of I such that 7r(Ao n I) = {O}. We may regard 7r as a
-representation of A. If we denote by (ei)i an approximate unit of I, then
we have 7r(a 0 ao) = lim 7r(a 0 eiao) = 0 for every ao E Ao, where the limit is
taken in the strong operator topology. This contradicts the fullness of Ao.
Pretty generous, eh?)
Exercise 4.6.2. Let Ao be a full, hereditary C -subalgebra of A. Prove
that A is nuclear (resp. exact) if and only if Ao is. (Another generous hint:
Let 7r: A -+ D be a -homomorphism. Prove that 7r is nuclear if 7r[Ao is.
To do this, let B be any C -algebra. Since Ao c A is hereditary, we have
Ao ®max B C A ®max B. Moreover, one checks that Ao ®max B is full and
hereditary. Let I= ker(A®maxB-+ A@B). Then, by the previous exercise,
ker(Ao ®max B -+ Ao ® B) =In (Ao ®max B)
is full in I. Therefore, the -homomorphism 7r ®max idB: A ®max B -+
D ®max B factors through A ® B, if one knows the same thing holds for
7r[Ao·)
Exercise 4.6.3. Let 'H be a Hilbert A-module. Prove that JK('H) is nuclear
(resp. exact) if A is. (Why stop now? Here is a proof: Let I= span{ \e, 17) :
e, 1J E 'H} c A. Since I is an ideal, I is nuclear (resp. exact) if A is. Hence,
we may assume that I = A, i.e., the Hilbert A-module 'H is full. It is not
hard to see that both JK('H) and A are full hereditary C -subalgebras of
JK('H EB A). Now apply the previous exercise.)
Exercise 4.6.4. Let 'H ® B be the exterior tensor product. Prove that the
natural inclusion IIB('H) 0 B <---+ IIB('H ® B) is min-continuous. (Hint: Realize
'H as a corner of IIB('H EB A).)