1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

336 11. Simple C* -Algebras

isometry with support el,1 and range nl,1; moreover, it can be shown that

llu1 - el,1ll2 is small. Now, defining Ui = ni,1u1e1,i, we get partial isometries

from ei,i to ni,i with the property that llui-ei,ill2 is small. Hence, u := l::i Ui

is the unitary we want. D

Lemma 11.5 .. 2. For any finite-dimensional subalgebra e in a II1 -factor M

and c; > 0 there exist an n E N and a unital inclusion M2n ( <C) C M such

that for each x Ee there exists y E M2n(<C) with llx -vll2:::; cllxll2·

Proof. We may assume lM E e. Let {Pih:s:;i:s:;k be the minimal projections

in the center of e (so e = p1e E9 · · · E9 Pke, where each Pie is a full matrix

algebra) and let {ep,q(i)} be matrix units for Pie. Assume first that there

is an n with the property that for each 1 :::; i :::; k there is an integer Si such

that 7M(e1,1(i)) = ;~. In this case standard manipulations, using the fact

that equivalence of projections is determined by the trace, produce a unital

inclusion M2n(<C) CM such that e C M2n(<C).

The general case is conceptually similar: Approximate the traces of the

ei,1 ( i)' s by appropriate dyadic rationals and then construct the approximat-

ing copy of M2n(<C) CM. D

If M happens to be AFD in the previous lemma, then an improve-

ment can be made: Any finite-dimensional subspace is almost contained in

M2n(<C) c M (since they can first be approximated by finite-dimensional

subalgebras in this case).

Lemma 11.5.3. Let R be a separable AFD II1-factor. For any finite-

dimensional subspace E c R, E > 0 and subfactor lR E N c R of type

I2n, there exists a sub factor M C R of type I2m (m > n) such that N C M

and for each x EE there exists y EM with llx - vll2:::; c;Jlxll ·

Proof. Using the previous lemma and the fact that R is AFD, we can find

some large m and a unital inclusion M2m (<C) C R such that both E and N

are nearly contained in M2m(<C) (in 2-norm). Now apply Lemma 11.5.1 to

find a unitary u E R which is close to 1 and such that N c uM 2 m (<C)u*. D

Theorem 11.5.4. Let R be a separable AFD II1 -factor and M 2 oo = M2(<C)0
M2 ( <C) 0 · · · be the UHF algebra of type 200 • Then R ~ 7r 7 ( M 2 oo )", where T
is the unique trace on M200 and 7r 7 is the GNS representation.
In particular, all separable injective II1 -factors are isomorphic.

Proof. Applying Lemma 11.5.3 over and over, it is easy to see that R
contains a weakly dense copy of M200. Since M2= has a unique tracial
state, uniqueness of GNS representations implies that R ~ 7r 7 (M 2 oo )", as
~~. D