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.

## Contents

* 1 For von Neumann algebras
* 2 Representations of C*-algebras
* 3 See also
* 4 References

## 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 PM and PN, onto M and N respectively, are elements of M, M « N if PM « PN.

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=M0⊃N0

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=M0⊃N0⊃M1.

By induction,

M=M0⊃N0⊃M1⊃N1⊃M2⊃N2⊃⋯.

It is clear that

R=∩i≥0Mi=∩i≥0Ni.

Let

M⊖N=defM∩(N)⊥.

M=⊕i≥0(Mi⊖Ni)⊕⊕j≥0(Nj⊖Mj+1)⊕R

and

N0=⊕i≥1(Mi⊖Ni)⊕⊕j≥0(Nj⊖Mj+1)⊕R.

Notice

Mi⊖Ni∼M⊖Nfor alli.

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

ρ=ρ1≃ρ1′⊕σ1whereσ1≃σ.

In turn,

ρ1≃ρ1′⊕(σ1′⊕ρ2)whereρ2≃ρ.

By induction,

ρ1≃ρ1′⊕σ1′⊕ρ2′⊕σ2′⋯≃(⊕i≥1ρi′)⊕(⊕i≥1σi′),

and

σ1≃σ1′⊕ρ2′⊕σ2′⋯≃(⊕i≥2ρi′)⊕(⊕i≥1σi′).

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

ρi′≃ρj′andσi′≃σj′for alli,j.

This proves the theorem.

## See also

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

## References

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

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