- Type I C* -algebras 55
2.7. Type I C*-algebras
We now consider the class of C* -algebras characterized by having nice rep-
resentation theory (in Glimm's sense) and which typically arise in (nondis-
crete) group representation theory. Our objective is to show that every
C*-algebra of type I is nuclear. In fact, nuclearity can be used to charac-
terize these algebras^19 but this will have to wait as it requires, among other
things, the nontrivial fact that nuclearity passes to quotients.
We begin with some von Neumann algebraic preliminaries. Recall that
every von Neumann algebra of type I is of the form
II A® JB(Hi)
iEJ
where I is a set of cardinal numbers, each Ai is an abelian von Neumann
algebra and Hi is a Hilbert space of dimension i (see Definition 1.3.1). Our
first task is to show that all of these algebras are semidiscrete.
Lemma 2.7.1. Let MC JB(H) be a van Neumann algebra and assume there
exists a net of projections P>.. EM such that P>.. --+ 11-l in the strong operator
topology and each of the corners P>..MP>.. is semidiscrete. Then M is also
semidiscrete.Proof. Given finite sets~ c M, n c 1-l and c; > O, we must show that there
exist c.c.p. maps cp: M --+ Mn(C) and 'I/;: Mn(C) --+ M such that
1(1/J o cp(m)~,'IJ) - (m~,'IJ)I < c;
for all m E ~and ~,'I] E 0 (see Remark 2.1.3). To do this, we first take
some projection p EM such that pMp is semidiscrete and llP~ -~II < c; for
all~ E 0. We then find c.c.p. maps rp: pMp-+ Mn(C) and 'lj;: Mn(C)--+ M
such that
1(1/J o rp(pmp)~, 'IJ) - (pmp~, 'IJ)I < c;
for all m E ~and~' 'IJ En. A standard bit of estimating shows that the c.c.p.
map cp: M--+ Mn(C) given by cp(m) = rp(pmp) together with 'I/; satisfies the
required inequality (with a slightly larger c;). D
Proposition 2.7.2. Every van Neumann algebra of type I is semidiscrete.Proof. First let us handle the special case A® JB(H), where A is abelian.
If 1-l happens to be finite dimensional, then A® lB ( 1-l) ~ Mn (A) is a nuclear
C* -algebra, by Corollary 2.4.4, and this evidently implies semidiscreteness.
For infinite-dimensional 1-l we first represent A® JB(H) c JB(JC@ 1-l), where
A c JB(JC) is any faithful normal representation. For a set n c 1-l we let
q 0 E JB(H) be the projection onto the span of n. Then we note that the net
19 A is type I if and only if every subalgebra of A is nuclear (Corollary 9.4.5).