Schröder–Bernstein property

Mirror-in-mirror images as counterexample: The left image can be embedded into the right one and vice versa (below, left/mid); yet, both aren't similar. The Schröder-Bernstein theorem applied to the unstructured pixel sets obtains a non-continuous bijection (right).

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.

The Schröder–Bernstein theorem for measurable spaces[1] states the Schröder–Bernstein property for the following case:

• 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,[2]

• 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,[3] 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.

The Myhill isomorphism theorem can be viewed as a Schröder–Bernstein theorem in computability theory.

Banach spaces violate the Schröder–Bernstein property;[4][5] here

• objects are Banach spaces,
• "a part" is interpreted as a subspace[4] or a complemented subspace,[5]
• "similar" is interpreted as linearly homeomorphic.

Many other Schröder–Bernstein problems related to various spaces and algebraic structures (groups, rings, fields etc.) are discussed by informal groups of mathematicians (see External Links below).

• Commutative von Neumann algebras

This article incorporates material from the Citizendium article "Schröder–Bernstein property", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.

• Srivastava, S.M. (1998), A Course on Borel Sets, Springer, ISBN 0-387-98412-7.
• Kadison, Richard V.; Ringrose, John R. (1986), Fundamentals of the theory of operator algebras, 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, doi:10.1112/blms/28.3.297, hdl:10338.dmlcz/127757.
• Casazza, P.G. (1989), "The Schroeder–Bernstein property for Banach spaces", Contemp. Math., 85: 61–78, doi:10.1090/conm/085/983381, MR 0983381.

• Theme and variations: Schroeder-Bernstein - Various Schröder–Bernstein problems are discussed in a group blog by 8 recent Berkeley mathematics Ph.D.
• When does Cantor Bernstein hold? - "Mathoverflow" discusses the question in terms of category theory: "Can we characterize Cantor-Bernsteiness in terms of other categorical properties?"