1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

11.4. Excision and Papa's technique 333

A II1-factor M is said to be approximately finite-dimensional ( AFD)

if the following holds: For every finite set ~ C M there exists a finite-

dimensional subalgebra B c M such that ~ C^5 '^2 B (i.e., for each x E ~

there exists b EB such that llx -bll2 < c:, where llYll~ = T(y*y) and Tis the

trace on M).

Lemma 11.4. 7. Let M be an injective II1 -factor. Then, for every finite set

~ C M and E > 0 ther:e exists a finite-dimensional subalgebra B c M with

unit q such that II [a, q] 112 < c:llqll2 and d2(qaq, B) < c:llqll2, for all a E ~.

Proof. Let M C JIB ( L^2 ( M)) be in standard form. Since there is a conditional
expectation JB(L^2 (M)) --+ M, the trace T on M is amenable (Definition
6.2.1). By Theorem 6.2.7 and Voiculescu's Theorem, there exists a net of
finite-rank projections Pi E JB(L^2 (M)) such that

and

ll[x,Pi]IJ2--+ O

11~112

Tr(xPi)
Tr(Pi) --+ T(x),
for all x E M (Exercise 6.2.5). Thus, as in the proof of the last theo-
rem, Papa's technique together with the commutator estimates finish off
the proof. D

Theorem 11.4.8. Injective II1 -factors are AFD.

Proof. Let M be an injective II1-factor. As is easily verified, the previous

lemma implies that M enjoys the following approximation property: For

each finite set ~ c M and E > 0 .there exists a finite-dimensional matrix

algebra B C M with unit q =/= 0 such that

llEB(qxq) - (x - qJ_xqJ_) II~~ c:^2 llqll~,

for all x E i, where EB: 'qM q --+ B is the unique trace-preserving conditional

expectation. It is important to note that every corner of M is also injective

(since x H pxp defines a conditional expectation M --+ pMp); hence it

enjoys the same approximation property.

Now fix a finite set .~ c M, E > 0 and consider the set S of all families

of matrix subalgebras {BihEI of M such that their units qi are mutually

orthogonal and

llEIJiBJqxq)-(x-qj_xqj_)ll~ ~ c:^2 llqll~,

where q = L:i qi. The set Sis nonempty, partially ordered by inclusion and

it is easily seen that every increasing sequence in S has a least upper bound

in S (by continuity, with respect to weak limits, of the quantities appearing