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.

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

${\displaystyle M=M_0\supset N_0}$

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

${\displaystyle M=M_0\supset N_0\supset M_1.}$

By induction,

${\displaystyle M=M_0\supset N_0\supset M_1\supset N_1\supset M_2\supset N_2\supset \cdots .}$

It is clear that

${\displaystyle R={\cap }_{i\ge 0}{M}_i={\cap }_{i\ge 0}{N}_i.}$

Let

${\displaystyle M\ominus N\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}M\cap (N{)}^{\perp }.}$

So

${\displaystyle M={\oplus }_{i\ge 0}(M_i\ominus N_i)\oplus {\oplus }_{j\ge 0}(N_j\ominus M_{j+1})\oplus R}$

and

${\displaystyle N_0={\oplus }_{i\ge 1}(M_i\ominus N_i)\oplus {\oplus }_{j\ge 0}(N_j\ominus M_{j+1})\oplus R.}$

Notice

${\displaystyle M_i\ominus N_i\sim M\ominus N\text{for all}i.}$

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

${\displaystyle \rho ={\rho }_1\simeq {{\rho }_1}^{\prime }\oplus {\sigma }_1\text{where}{\sigma }_1\simeq \sigma .}$

In turn,

${\displaystyle {\rho }_1\simeq {{\rho }_1}^{\prime }\oplus ({{\sigma }_1}^{\prime }\oplus {\rho }_2)\text{where}{\rho }_2\simeq \rho .}$

By induction,

${\displaystyle {\rho }_1\simeq {{\rho }_1}^{\prime }\oplus {{\sigma }_1}^{\prime }\oplus {{\rho }_2}^{\prime }\oplus {{\sigma }_2}^{\prime }\cdots \simeq ({\oplus }_{i\ge 1}{{\rho }_i}^{\prime })\oplus ({\oplus }_{i\ge 1}{{\sigma }_i}^{\prime }),}$

and

${\displaystyle {\sigma }_1\simeq {{\sigma }_1}^{\prime }\oplus {{\rho }_2}^{\prime }\oplus {{\sigma }_2}^{\prime }\cdots \simeq ({\oplus }_{i\ge 2}{{\rho }_i}^{\prime })\oplus ({\oplus }_{i\ge 1}{{\sigma }_i}^{\prime }).}$

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

${\displaystyle {{\rho }_i}^{\prime }\simeq {{\rho }_j}^{\prime }\text{and}{{\sigma }_i}^{\prime }\simeq {{\sigma }_j}^{\prime }\text{for all}i,j.}$

This proves the theorem.

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

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

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