Schröder–Bernstein theorems for operator algebras

For von Neumann algebras

Suppose M is a von Neumann algebra and E, F are projections in M. Let ~ denote the Murray-von Neumann equivalence relation on M. Define a partial order « on the family of projections by E « F if E ~ F' ≤ F. In other words, E « F if there exists a partial isometry U ∈ M such that U*U = E and UU* ≤ F.

For closed subspaces M and N where projections P_M and P_N, onto M and N respectively, are elements of M, M « N if P_M « P_N.

The Schröder–Bernstein theorem states that if M « N and N « M, then M ~ N.

A proof, one that is similar to a set-theoretic argument, can be sketched as follows. Colloquially, N « M means that N can be isometrically embedded in M. So

M = M 0 ⊃ N 0

where N₀ is an isometric copy of N in M. By assumption, it is also true that, N, therefore N₀, contains an isometric copy M₁ of M. Therefore one can write

M = M 0 ⊃ N 0 ⊃ M 1 .

By induction,

M = M 0 ⊃ N 0 ⊃ M 1 ⊃ N 1 ⊃ M 2 ⊃ N 2 ⊃ ⋯ .

It is clear that

R = ∩ i ≥ 0 M i = ∩ i ≥ 0 N i .

Let

M ⊖ N = d e f M ∩ ( N ) ⊥ .

So

M = ⊕ i ≥ 0 ( M i ⊖ N i ) ⊕ ⊕ j ≥ 0 ( N j ⊖ M j + 1 ) ⊕ R

and

N 0 = ⊕ i ≥ 1 ( M i ⊖ N i ) ⊕ ⊕ j ≥ 0 ( N j ⊖ M j + 1 ) ⊕ R .

Notice

M i ⊖ N i ∼ M ⊖ N for all i .

The theorem now follows from the countable additivity of ~.

Representations of C*-algebras

There is also an analog of Schröder–Bernstein for representations of C*-algebras. If A is a C*-algebra, a representation of A is a *-homomorphism φ from A into L(H), the bounded operators on some Hilbert space H.

If there exists a projection P in L(H) where P φ(a) = φ(a) P for every a in A, then a subrepresentation σ of φ can be defined in a natural way: σ(a) is φ(a) restricted to the range of P. So φ then can be expressed as a direct sum of two subrepresentations φ = φ' ⊕ σ.

Two representations φ₁ and φ₂, on H₁ and H₂ respectively, are said to be unitarily equivalent if there exists an unitary operator U: H₂ → H₁ such that φ₁(a)U = Uφ₂(a), for every a.

In this setting, the Schröder–Bernstein theorem reads:

If two representations ρ and σ, on Hilbert spaces H and G respectively, are each unitarily equivalent to a subrepresentation of the other, then they are unitarily equivalent.

A proof that resembles the previous argument can be outlined. The assumption implies that there exist surjective partial isometries from H to G and from G to H. Fix two such partial isometries for the argument. One has

ρ = ρ 1 ≃ ρ 1 ′ ⊕ σ 1 where σ 1 ≃ σ .

In turn,

ρ 1 ≃ ρ 1 ′ ⊕ ( σ 1 ′ ⊕ ρ 2 ) where ρ 2 ≃ ρ .

By induction,

ρ 1 ≃ ρ 1 ′ ⊕ σ 1 ′ ⊕ ρ 2 ′ ⊕ σ 2 ′ ⋯ ≃ ( ⊕ i ≥ 1 ρ i ′ ) ⊕ ( ⊕ i ≥ 1 σ i ′ ) ,

and

σ 1 ≃ σ 1 ′ ⊕ ρ 2 ′ ⊕ σ 2 ′ ⋯ ≃ ( ⊕ i ≥ 2 ρ i ′ ) ⊕ ( ⊕ i ≥ 1 σ i ′ ) .

Now each additional summand in the direct sum expression is obtained using one of the two fixed partial isometries, so

ρ i ′ ≃ ρ j ′ and σ i ′ ≃ σ j ′ for all i , j .

This proves the theorem.