1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

F. Bimodules over von Neumann Algebras 487

Wn E En such that llEAn (xjwnxi) 112 :S n-^1 for every i, j :S n. Let B1 =

ClN and suppose we have constructed B1, · · · , En. Let {pk} be the set

of minimal projections in En. Since PkN Pk does not satisfy condition (1)

in Theorem F.12, there exists a unitary element Vk E PkNPk such that

llEAn+i (xjvkxi)ll2 :S ((n + 1) dimBn)-^1 for every i,j :S n + 1. We may

assume that every Vk has finite spectrum. It is easy to see that Wn+i =:Ek vk

and the finite-dimensional abelian subalgebra Bn+l generated by En and

Wn+i satisfy the required property. Since limn llEAn(xjwnxi)ll2 = 0 for

every i,j, the abelian von Neumann subalgebra B = V En C N does not

satisfy condition (1) for any An. Finally, it is clear that B is diffuse. D

There are several instances in which local embeddability implies global

embeddability. The first example involves the notion of a Cartan subalgebra.

Definition F.15. Let M be a II1-factor. A Cartan subalgebra is a maximal

abelian von Neumann subalgebra Ac M such that the normalizer

N(A) = {u EM: a unitary element such that uAu* =A}

generates Mas a von Neumann algebra.

Lemma F.16. Let Ac M be a Cartan subalgebra of a Ih-factor. If pro-

jections e, f E A have the same trace, then there is u E N(A) such that

ueu* = f.

Proof. Let ( { ei}, { Ui}) be a maximal pair of sequences of nonzero pro-
jections ei E A and unitary elements ui E N(A) such that I: ei :S e and
I: Uieiui :S f. We claim that e' := e-I: ei = 0. Indeed, if e^1 is nonzero, then
f' = f -I: Uieiui is also nonzero since it has the same trace as e'. Since the
projection VuEN(A) u f'u EM commutes with N(A), it has to be 1 by fac-
toriality of M. Therefore, there is uo E N(A) such that eo := uOf'uoe' =/::. 0.
The pair (ea, uo) contradicts the maximality of ( { ei}, { Ui}); hence e = I: ei,
as claimed.
Thus v = I: Uiei E M is a partial isometry such that vv = e, vv = f
and vAv = Af. Applying the same argument to e..l and f ..l, we obtain a
partial isometry w E M such that ww = e..l, ww = f ..l and wAw* = Af ..L.
Setting u = v + w, we are done. D

Lemma F.17. Let A and B be Cartan subalgebras of a II1 -factor M. If
B embeds in A inside M, then A and B are unitarily conjugate, i.e., there
exists a unitary element u E M such that uAu* = B.

Proof. Let e EA, f EB, 0: f Bf----+ eAe and v EM be as in condition (4)
of Theorem F.12:

Vx E fBf, xv= vO(x) and vv E O(f Bf)' n eMe, vv E (B' n M)f.