1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

40 2. Nuclear and Exact C*-Algebras

and where P E Mn(Co(X)) is a projection. Hence various exercises imply

this corollary.^10 D

Remark 2.4.6. It follows from the corollary above and some work of Elliott

and Gong that for any countable dimension group Go and any other count-

able abelian group G1 we can find an AH algebra A such that Ko(A) ~Go

and K 1 (A) c;:,,!. G 1 (cf. [ 65]). In fact their result is much sharper, but our point

is that we have shown the existence of a large class of nuclear C* -algebras.

As might be expected, it turns out our C* -notions impose tremendous

restrictions on von Neumann algebras. Since we haven't yet demonstrated

the existence of nonexact C* -algebras, a tiny bit of faith will be required of

the reader.

Lemma 2.4.'7. Let k(n) be a sequence of integers tending to infinity. Then

the van Neumann algebra.

M =IT Mk(n)(C) = {(xn): Xn E Mk(n)(C), sup Jlxnll < oo}

nEN n

is not exact.

Proof. Eventually we'll prove that there exists a separable nonexact C* -

algebra A which has a sequence of representations 1fn: A ----> Mj(n) (C) such

that EBn 1fn is faithful; see Corollary 3.7.12 and.Theorem 7.4.1. With this in

hand, the remainder of the proof is simple. Indeed, the fact that k(n) ----> oo

ensures that there is enough room to embed A into M by putting a separat-

ing family of finite-dimensional representations into the tail of M (i.e., for a

given representation 1f: A ----> Mj ( q we simply take a nonunital embedding

Mj(C) C Mk(n)(C), for sufficiently large n). Hence M contains a nonexact

subalgebra and thus is not exact (since exactness passes to subalgebras). D

Recall that a projection p E M is called abelian if pMp is an abelian

algebra. It is a fact that no von Neumann algebra of type II or III contains

an abelian projection (this is required in the proof of the type decomposition

explained in Section 1.3).

Lemma 2.4.8. Assume a van Neumann algebra M has no abelian projec-

tions. Then for every n EN there is an embedding Mn(C) '---+ M.

Proof. It suffices to show the existence of pairwise orthogonal projections
Pl, ... , Pn E M with the property that there exist partial isometries v1, ... , Vn
such that vivi = Pl and Vivi = Pi for i = 1, ... , n (i.e., Pi rv Pj for
all i,j); defining ei,j = Vivj', we get a system of matrix units and hence
C*({ei,j: 1 :'S i,j :'Sn})~ Mn(C).

(^10) In case one hasn't studied these things before, PMn(Go(X))P is a hereditary subalgebra
of Mn(Go(X)).