11.5. Connes's uniqueness 'theorem 33511.5. Connes's uniqueness theorem
We close this chapter with the celebrated uniqueness theorem for injective
II1-factors. Having reduced the problem to the AFD case in the previous
section, we only have to present Murray and von Neumann's uniqueness
theorem for AFD II1-factors. The main ingredients in the proof are some
nontrivial finite-dimensional perturbation lemmas. However, these prelimi-
naries are well known and readily accessible in existing books ([185, Lemmas
XIV.2.8 and XIV.2.10] or [95, Lemmas 12.2.3 - 12.2.6]), so we only sketch
the main points. (The omitted details also make very good exercises.)Lemma 11.5.1. Let Mn(<C) C M be a unital inclusion, where M is a II1-
f actor, and let { ei,j} be matrix units for Mn ( <C). For every E > 0 there exists
a o = o(n, c:) > 0 with the following property: If lM EN c M is a subfactor
of type Inp (i.e., N = Mnp(<C)) with the property that for each i,j there is a
contraction ni,j EN such that llei,j - ni,j 112 < o, then there exists a unitary
u EM such that llu - lll2 < E and Mn(<C) Cu* Nu.Proof. The first step is to prove a version of this for orthogonal projections.
More precisely, one shows that if p, q E M are orthogonal projections, v E M
is a partial isometry with support p and range q, and N c M is a type I
subfactor which almost contains p, q and v (in 2-norm), then there is a partial
isometry s EN such that llss - Pll2, llss - qll2 and lls - vll2 are all small.
With the projection version in hand one constructs matrices. That is,
under the hypotheses of the lemma, we can recursively apply the projection
result to find orthogonal projections Fi,i E N (1 ::::; i ::::; n) and partial
isometries Vi+i,i EN such that llFi,i - ei,ill2 and llVi+i,i - ei+l,ill2 are small
and Vi+i,i has support Fi,i and range Fi:+i,i+l· Sadly, the nxn matrix algebra
generated by the Fi,i's and Vi+i,i's need not contain the unit of M, but this
is easy to fix. Indeed, since N is of type Inp and T(Fi,i) = T(Fj,j) = !:,P
for some k ::::; p, it follows that T(l - :Ei Fi,i) = n(~;k); hence, 1 - :Ei Fi,i
is also the sum of n orthogonal, mutually equivalent projections (of trace
P;;:). Adding these projections and the corresponding partial isometries to
the Fi,i's and Vi+i/s gives a unital n x n matrix subalgebra of N which
nearly contains the original copy of Mn ( <C) C M.
The upshot of the previous paragraph is that when attacking the general
case, we may assume that N is also an n x n matrix algebra with matrix
units { ni,j} that are close to the matrix units { ei,j}. Constructing the right
unitary is again piecemeal, via projection considerations. To start, if v1,1 is
the partial isometry in the polar decomposition of n1,1 e1,1 and w1,1 is any
partial isometry with support e1,1 - vi, 1 v1,1 and range n1,1 - v1,1 vi, 1 (which
exists, since e1,1 and n1,1 are equivalent), then u1 := v1,1 + w1,1 is a partial