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 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 \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 }}.

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 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

\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.

References

B. Blackadar, Operator Algebras, Springer, 2006.

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Schröder–Bernstein theorems for operator algebras

For von Neumann algebras

Representations of C*-algebras

See also

References