Schröder–Bernstein Theorem For Operator Algebras
Schröder–Bernstein may refer to: *the Schröder–Bernstein theorem in set theory *Schröder–Bernstein theorem for measurable spaces *Schröder–Bernstein theorems for operator algebras *Schröder–Bernstein property A Schröder–Bernstein property is any 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 ...

Schröder–Bernstein Theorem
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the sets and , then there exists a bijective function . In terms of the cardinality of the two sets, this classically implies that if and , then ; that is, and are equipotent. This is a useful feature in the ordering of cardinal numbers. The theorem is named after Felix Bernstein and Ernst Schröder. It is also known as Cantor–Bernstein theorem, or Cantor–Schröder–Bernstein, after Georg Cantor who first published it without proof. Proof The following proof is attributed to Julius König. Assume without loss of generality that ''A'' and ''B'' are disjoint. For any ''a'' in ''A'' or ''b'' in ''B'' we can form a unique two-sided sequence of elements that are alternately in ''A'' and ''B'', by repeatedly applying f and g^ to go from ''A'' to ''B'' and g and f^ to go from ''B'' to ''A'' (where defined; the inverses f^ and g^ are understood as partial functi ...
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Schröder–Bernstein Theorem For Measurable Spaces
The Cantor–Bernstein–Schroeder theorem of set theory has a counterpart for measurable spaces, sometimes called the Borel Schroeder–Bernstein theorem, since measurable spaces are also called Borel spaces. This theorem, whose proof is quite easy, is instrumental when proving that two measurable spaces are isomorphic. The general theory of standard Borel spaces contains very strong results about isomorphic measurable spaces, see Kuratowski's theorem. However, (a) the latter theorem is very difficult to prove, (b) the former theorem is satisfactory in many important cases (see Examples), and (c) the former theorem is used in the proof of the latter theorem. The theorem Let X and Y be measurable spaces. If there exist injective, bimeasurable maps f : X \to Y, g : Y \to X, then X and Y are isomorphic (the Schröder–Bernstein property). Comments The phrase "f is bimeasurable" means that, first, f is measurable (that is, the preimage f^(B) is measurable for every measurable B ...
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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. For von Neumann algebras Suppose M is a von Neumann algebra and ''E'', ''F'' are projections in M. Let ~ denote the Von Neumann algebra#Projections, 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 isometric ...
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