Disintegration theory for von Neumann algebras

2.4 Decomposable von Neumann algebras
be a separable, strong operator dense C ∗ -algebra in M. That this is always the case is the proved in
Proposition 2.30 and is interesting in its own right. The other is that the decomposition of M might
depend on the choice of the subalgebra A. Lemma 2.31 takes care of this.
Lemma 2.29. If ϕ : M → N is a ∗ -isomorphism between von Neumann algebras, and A is a
strong operator dense, norm separable C ∗ -subalgebra of M, then ϕ(A) is a strong operator dense,
separable C ∗ -subalgebra of N.
Proof. The lemma is a consequence Kaplansky’s density theorem and the fact that ∗ -isomorphisms
are strong operator continuous on bounded sets. Details are left to the reader.
□
Proposition 2.30. A von Neumann algebra M acting on a separable Hilbert space H has a strong
operator dense, norm separable C ∗ -subalgebra.
Proof. Let ι : M → B(H) denote the identity representation, and let {x n } be a countable, dense
subset of H consisting of non-zero vectors. The inflation of ι, ι (∞) = ⊕ ∞
n=1
ι is a faithful representation
of M on the separable Hilbert space ⊕ ∞
n=1 H, and the image ι(∞) (M) has a separating vector,
(||x 1 || −1 x 1 , ||x 2 || −2 x 2 , . . .). Thus, by the previous lemma we may assume that M has a separating
vector.
Then every normal functional on M is of the form T ↦→ ω x,y (T) = 〈T x, y〉 (see for instance Theorem
7.3.3 in [K&R II]). In the following ||ω x,y || denotes the norm of the functional viewed as a functional
on M (as opposed to a functional on all of B(H)). For each m, n ∈ N choose A mn in (M) 1 such that
ω xm ,x m
(A mn ) ≥ ||ω xm ,x n
|| − 1 4 , and let A be the C∗ -algebra generated by {A mn }. Notice that A is norm
separable. We prove that A is strong operator dense in M.
Suppose ||ω x,y |A|| = 0, but ||ω x,y || = 1 for some x, y ∈ H. Let ε > 0 be given, and choose x m , x n such
that ||x m − x|| and ||x n − y|| are less that ε. Then for ||T|| ≤ 1
|ω x,y (T) − ω xm ,x n
(T)| = |〈T x, y〉 − 〈T x m , x n 〉|
= |〈T x, y − x n 〉 + 〈T(x − x m ), x n 〉|
≤ ||T|| ||x|| ||y − x n || + ||T|| ||x − x m || ||x n ||
≤
(||x|| + ||y|| + ε)ε.
So if we let ε be such that (||x|| + ||y|| + ε)ε < 1 4 , then ||ω x,y − ω xm ,x n
|| < 1 4 .
But also
||ω x,y − ω xm ,x n
|| ≥ |(ω x,y − ω xm ,x n
)(A mn )| = ω xm ,x n
(A mn )
≥ ||ω xm ,x n
|| − 1 4 ≥ ||ω x,y|| − 1 2 = 1 2 .
This gives a contradiction, and so each normal functional annihilating A also annihilates M.
Since the normal functionals are the ultraweakly continuous functionals, we get by the Hahn-Banach
theorem ([K&R I] theorem 1.2.13), that the closure of A in the ultraweak topology is M.
20