1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

15.3. Examples 419

of A, we can apply Lemma 5.2.8 and deduce that snrio -+ z. Hence we

obtain the contradiction

c: ~ lim n lf(sno) - f(snCio)I = lf(z) - f(z)I = 0.

Therefore, O"(j) - O"(j)t E c 0 (r; {A}) c c 0 (r; Q) and we are done. D

Theorem 15.3.13. Let I'= I'i *A I'2 be an amalgamated free product such

that both ri are exact and A is amenable. If N c L(r) is a von Neumann

subalgebra with a noninjective relative commutant, then there exists i such

that N embeds in L(I'i) inside L(I').

Proof. Combine Theorem 15.1.5 and Proposition 15.3.12. D

Recall that a group r is said to have infinite conjugacy classes (ICC) if

the sets { sts-i : s E r} are infinite for every nonneutral element t E r.

Corollary 15.3.14. Let r = I'i * I'2 be a free product of ICC exact groups.

If N C L(I') is a noninjective nonprime factor whose relative commutant

N' n L(r) is a factor, then there exist i E {1, 2} and a unitary element

u E L(r) such that uNu* c L(ri)·

We omit the proof of this corollary as it is very similar to the proof of
Corollary 15.3.11.
We say r is a product group if it is isomorphic to a direct product of
nontrivial groups. We note that if r = r' x r" is an ICC product group,
then r' and r" are also ICC and, in particular, infinite.

Corollary 15.3.15. Let I'i, ... , r n and Ai, ... , Am be ICC nonamenable

exact product groups. If

M = L(lFoo * I'i * · · · * r n) ~ L(lFoo *Ai*···* Am),

then n = m and, modulo permutation of indices, L(I'i) is unitarily conju-

gated to L(Ai) inside M for every 1 ~ i ~ n.

Proof. It follows from Corollary 15.3.14 that there are maps i, J and uni-
tary elements ui, ... , Um and vi, ... Vn such that UjL(Aj)uj C L(I'i(j)) and
viL(I'i)vi c L(AJ(i))· It follows that
Vi(j)UjL(Aj)ujv;(j) C L(AJ(i(j)))

for every j. By Theorem F.20 and Exercise F.8, this implies J(i(j)) =
j and Vi(j)Uj E L(Aj)· In particular, the above inclusions are tight and
UjL(Aj)uj = L(I'i(j))· Likewise, one has i(J(i)) = i for every 1 ~ i ~ n. D

This corollary is an analogue of Kurosh's isomorphism theorem for groups
and, like Kurosh's Theorem, it says almost nothing about the positions of
the copies of L(JF 00 ).