Schroeder-Bernstein property

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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).

Schröder-Bernstein properties

In order to define a specific Schröder-Bernstein property one should decide

In the classical (Cantor-)Schröder–Bernstein theorem,

Not all statements of this form are true. For example, assume that

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

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:

In the Schröder–Bernstein theorem for operator algebras,[2]

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.

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

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

Notes

  1. Srivastava 1998, see Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).
  2. Kadison & Ringrose 1986, see Proposition 6.2.4 (on page 406).
  3. Kadison & Ringrose 1986, see Theorem 9.4.1 (on page 666).
  4. 4.0 4.1 Casazza 1989
  5. 5.0 5.1 Gowers 1996

References

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