Schröder–Bernstein theorems for operator algebras
# Schröder–Bernstein theorems for operator algebras

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 $N_0$ is an isometric copy of N in M. By assumption, it is also true that, N, therefore $N_0,$ contains an isometric copy $M_1$ of M. Therefore, one can write

    $M=M_{0}\supset N_{0}\supset M_{1}.$

By induction,

    $M=M_{0}\supset N_{0}\supset M_{1}\supset N_{1}\supset M_{2}\supset N_{2}\supset \cdots .$

It is clear that

    $R=\cap _{i\geq 0}M_{i}=\cap _{i\geq 0}N_{i}.$

Let

    $M\ominus N{\stackrel {\mathrm {def} }{=}}M\cap (N)^{\perp }.$

So

    $M=\oplus _{i\geq 0}(M_{i}\ominus N_{i})\quad \oplus \quad \oplus _{j\geq 0}(N_{j}\ominus M_{j+1})\quad \oplus R$

and

    $N_{0}=\oplus _{i\geq 1}(M_{i}\ominus N_{i})\quad \oplus \quad \oplus _{j\geq 0}(N_{j}\ominus M_{j+1})\quad \oplus R.$

Notice

    $M_{i}\ominus N_{i}\sim M\ominus N\quad {\mbox{for all}}\quad i.$

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 $φ_1$ and $φ_2,$ on $H_1$ and $H_2$ respectively, are said to be unitarily equivalent if there exists a unitary operator U: $H_2$ → $H_1$ such that $φ_1(a)U$ = $Uφ_2(a),$ 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

    $\rho =\rho _{1}\simeq \rho _{1}'\oplus \sigma _{1}\quad {\mbox{where}}\quad \sigma _{1}\simeq \sigma .$

In turn,

    $\rho _{1}\simeq \rho _{1}'\oplus (\sigma _{1}'\oplus \rho _{2})\quad {\mbox{where}}\quad \rho _{2}\simeq \rho .$

By induction,

    $\rho _{1}\simeq \rho _{1}'\oplus \sigma _{1}'\oplus \rho _{2}'\oplus \sigma _{2}'\cdots \simeq (\oplus _{i\geq 1}\rho _{i}')\oplus (\oplus _{i\geq 1}\sigma _{i}'),$

and

    $\sigma _{1}\simeq \sigma _{1}'\oplus \rho _{2}'\oplus \sigma _{2}'\cdots \simeq (\oplus _{i\geq 2}\rho _{i}')\oplus (\oplus _{i\geq 1}\sigma _{i}').$

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

    $\rho _{i}'\simeq \rho _{j}'\quad {\mbox{and}}\quad \sigma _{i}'\simeq \sigma _{j}'\quad {\mbox{for all}}\quad i,j\;.$

This proves the theorem.

