The Schröder–Bernstein theorem from set theory has analogs in the context of 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 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 = M 0 ⊃ N 0 {\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
M = M 0 ⊃ N 0 ⊃ M 1 . {\displaystyle M=M_{0}\supset N_{0}\supset M_{1}.}
By induction,
M = M 0 ⊃ N 0 ⊃ M 1 ⊃ N 1 ⊃ M 2 ⊃ N 2 ⊃ ⋯ . {\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
R = ∩ i ≥ 0 M i = ∩ i ≥ 0 N i . {\displaystyle R=\cap _{i\geq 0}M_{i}=\cap _{i\geq 0}N_{i}.}
Let
M ⊖ N = d e f M ∩ ( N ) ⊥ . {\displaystyle M\ominus N{\stackrel {\mathrm {def} }{=}}M\cap (N)^{\perp }.}
So
M = ⊕ i ≥ 0 ( M i ⊖ N i ) ⊕ ⊕ j ≥ 0 ( N j ⊖ M j + 1 ) ⊕ R {\displaystyle 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 = ⊕ i ≥ 1 ( M i ⊖ N i ) ⊕ ⊕ j ≥ 0 ( N j ⊖ M j + 1 ) ⊕ R . {\displaystyle 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 ⊖ N i ∼ M ⊖ N for all i . {\displaystyle 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 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
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