A mathematical property that matches the following pattern
- If, for some mathematical objects X and Y, both X is similar to a part of Y and Y is similar to a part of X then X and Y are similar (to each other).
is often called a Schröder–Bernstein (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) property in analogy to the theorem of the same name (from set theory).
In order to define a specific Schröder-Bernstein property one should decide
- what kind of mathematical objects are X and Y,
- what is meant by "a part",
- what is meant by "similar".
In the classical (Cantor-)Schröder–Bernstein theorem,
- objects are sets (maybe infinite),
- "a part" is interpreted as a subset,
- "similar" is interpreted as equinumerous.
Not all statements of this form are true. For example, assume that
- objects are triangles,
- "a part" means a triangle inside the given triangle,
- "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles").
Then the statement fails badly: every triangle X evidently is similar to some triangle inside Y, and the other way round; however, X and Y need not be similar.
A Schröder–Bernstein property is a joint property of
- a class of objects,
- a binary relation "be a part of",
- a binary relation "be similar to" (similarity).
Instead of the relation "be a part of" one may use a binary relation "be embeddable into" (embeddability) interpreted as "be similar to some part of". Then a Schröder–Bernstein property takes the following form.
- If X is embeddable into Y and Y is embeddable into X then X and Y are similar.
The same in the language of category theory:
- If objects X, Y are such that X injects into Y (more formally, there exists a monomorphism from X to Y) and also Y injects into X then X and Y are isomorphic (more formally, there exists an isomorphism from X to Y).
The relation "injects into" is a preorder (that is, a reflexive and transitive relation), and "be isomorphic" is an equivalence relation. Also embeddability is usually a preorder, and similarity is usually an equivalence relation (which is natural, but not provable in the absence of formal definitions). Generally, a preorder leads to an equivalence relation and a partial order between the corresponding equivalence classes. The Schröder–Bernstein property claims that the embeddability preorder (assuming that it is a preorder) leads to the similarity equivalence relation, and a partial order (not just preorder) between classes of similar objects.
The problem of deciding whether a Schröder–Bernstein property (for a given class and two relations) holds or not, is called a Schröder–Bernstein problem. A theorem that states a Schröder–Bernstein property (for a given class and two relations), thus solving the Schröder–Bernstein problem in the affirmative, is called a Schröder–Bernstein theorem (for the given class and two relations), not to be confused with the classical (Cantor–)Schröder–Bernstein theorem mentioned above.
- objects are measurable spaces,
- "a part" is interpreted as a measurable subset treated as a measurable space,
- "similar" is interpreted as isomorphic.
In the Schröder–Bernstein theorem for operator algebras,
- objects are projections in a given von Neumann algebra;
- "a part" is interpreted as a subprojection (that is, E is a part of F if F – E is a projection);
- "E is similar to F" means that E and F are the initial and final projections of some partial isometry in the algebra (that is, E = V*V and F = VV* for some V in the algebra).
Taking into account that commutative von Neumann algebras are closely related to measurable spaces, one may say that the Schröder–Bernstein theorem for operator algebras is in some sense a noncommutative counterpart of the Schröder–Bernstein theorem for measurable spaces.
- objects are Banach spaces,
- "a part" is interpreted as a subspace or a complemented subspace,
- "similar" is interpreted as linearly homeomorphic.
- Srivastava, S.M. (1998), A Course on Borel Sets, Springer, ISBN 0387984127.
- Kadison, Richard V. & John R. Ringrose (1986), Fundamentals of the theory of operator algebras, vol. II, Academic Press, ISBN 0-12-393302-1.
- Gowers, W.T. (1996), "A solution to the Schroeder-Bernstein problem for Banach spaces", Bull. London Math. Soc. 28: 297–304.
- Casazza, P.G. (1989), "The Schroeder-Bernstein property for Banach spaces", Contemp. Math. 85: 61–78.