Schröder–Bernstein theorems for operator 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

:$M\; =\; M\_0\; \backslash supset\; N\_0$

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

:$M\; =\; M\_0\; \backslash supset\; N\_0\; \backslash supset\; M\_1.$

By induction,

:$M\; =\; M\_0\; \backslash supset\; N\_0\; \backslash supset\; M\_1\; \backslash supset\; N\_1\; \backslash supset\; M\_2\; \backslash supset\; N\_2\; \backslash supset\; \backslash cdots\; .$

It is clear that

:$R\; =\; \backslash cap\_\{i\; \backslash geq\; 0\}\; M\_i\; =\; \backslash cap\_\{i\; \backslash geq\; 0\}\; N\_i.$

Let

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

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

:$M\_i\; \backslash ominus\; N\_i\; \backslash sim\; M\; \backslash ominus\; N\; \backslash quad\; \backslash mbox\{for\; all\}\; \backslash quad\; i.$

The theorem now follows from the countable additivity of ~.

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

:$\backslash rho\; =\; \backslash rho\_1\; \backslash simeq\; \backslash rho\_1\; \text{'}\; \backslash oplus\; \backslash sigma\_1\; \backslash quad\; \backslash mbox\{where\}\; \backslash quad\; \backslash sigma\_1\; \backslash simeq\; \backslash sigma.$

In turn,

:$\backslash rho\_1\; \backslash simeq\; \backslash rho\_1\; \text{'}\; \backslash oplus\; (\backslash sigma\_1\; \text{'}\; \backslash oplus\; \backslash rho\_2)\; \backslash quad\; \backslash mbox\{where\}\; \backslash quad\; \backslash rho\_2\; \backslash simeq\; \backslash rho\; .$

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.

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

{{reflist}}

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

{{Spectral theory}}

{{DEFAULTSORT:Schroder-Bernstein theorems for operator algebras}}

Category:C*-algebras

Category:Theorems in functional analysis

Category:Operator theory

Category:Von Neumann algebras