Schröder–Bernstein theorems for operator algebras explained

The Schröder–Bernstein theorem from set theory has analogs in the context operator algebras. This article discusses such operator-algebraic results.

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} \supset 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} \supset N _{0} \supset M _{1.}

By induction,

M = M _{0} \supset N _{0} \supset M _{1} \supset N _{1} \supset M _{2} \supset N _{2} \supset … .

It is clear that

R = \cap _{i} M _{i} = \cap _{i} N _{i.}

Let

M \ominus N \stackrel{ def } M \cap (N)^.

So

M = ⊕ _{i} ( M _{i} \ominus N _{i} ) ⊕ ⊕ _{j} ( N _{j} \ominus M _{j+1} ) ⊕ R

and

N _{0} = ⊕ _{i} ( M _{i} \ominus N _{i} ) ⊕ ⊕ _{j} ( N _{j} \ominus M _{j+1} ) ⊕ R.

Notice

M _{i} \ominus N _{i} \sim M \ominus N forall 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 a 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

\rho = \rho _{1} \simeq \rho _{1} ' ⊕ \sigma _{1} where \sigma _{1} \simeq \sigma.

In turn,

\rho _{1} \simeq \rho _{1} ' ⊕ (\sigma _{1} ' ⊕ \rho _{2)} where \rho _{2} \simeq \rho .

By induction,

\rho _{1} \simeq \rho _{1} ' ⊕ \sigma _{1} ' ⊕ \rho _{2'} ⊕ \sigma _{2} ' … \simeq ( ⊕ _{i} \rho _{i} ' ) ⊕ ( ⊕ _{i} \sigma _{i} '),

and

\sigma _{1} \simeq \sigma _{1} ' ⊕ \rho _{2'} ⊕ \sigma _{2} ' … \simeq ( ⊕ _{i} \rho _{i} ' ) ⊕ ( ⊕ _{i} \sigma _{i} ').

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

\rho _{i} ' \simeq \rho _{j} ' and \sigma _{i} ' \simeq \sigma _{j} ' forall i,j .

This proves the theorem.

See also

• Schröder–Bernstein theorem for measurable spaces
• Schröder–Bernstein property

References

• B. Blackadar, Operator Algebras, Springer, 2006.