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The
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 ...
from
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
has analogs in the context of
operator algebras In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study o ...
. This article discusses such operator-algebraic results.


For von Neumann algebras

Suppose M is a
von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann al ...
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 \supset N_0 where ''N''0 is an isometric copy of ''N'' in ''M''. By assumption, it is also true that, ''N'', therefore ''N''0, contains an isometric copy ''M''1 of ''M''. Therefore, one can write :M = M_0 \supset N_0 \supset M_1. By induction, :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 = \cap_ M_i = \cap_ N_i. Let :M \ominus N \stackrel M \cap (N)^. So : M = \oplus_ ( M_i \ominus N_i ) \quad \oplus \quad \oplus_ ( N_j \ominus M_) \quad \oplus R and : N_0 = \oplus_ ( M_i \ominus N_i ) \quad \oplus \quad \oplus_ ( N_j \ominus M_) \quad \oplus R. Notice :M_i \ominus N_i \sim M \ominus N \quad \mbox \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 In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
. If ''A'' is a C*-algebra, a
representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
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 ''φ''1 and ''φ''2, on ''H''1 and ''H''2 respectively, are said to be unitarily equivalent if there exists a unitary operator ''U'': ''H''2 → ''H''1 such that ''φ''1(''a'')''U'' = ''Uφ''2(''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 :\rho = \rho_1 \simeq \rho_1 ' \oplus \sigma_1 \quad \mbox \quad \sigma_1 \simeq \sigma. In turn, :\rho_1 \simeq \rho_1 ' \oplus (\sigma_1 ' \oplus \rho_2) \quad \mbox \quad \rho_2 \simeq \rho . By induction, : \rho_1 \simeq \rho_1 ' \oplus \sigma_1 ' \oplus \rho_2' \oplus \sigma_2 ' \cdots \simeq ( \oplus_ \rho_i ' ) \oplus ( \oplus_ \sigma_i '), and : \sigma_1 \simeq \sigma_1 ' \oplus \rho_2' \oplus \sigma_2 ' \cdots \simeq ( \oplus_ \rho_i ' ) \oplus ( \oplus_ \sigma_i '). Now each additional summand in the direct sum expression is obtained using one of the two fixed partial isometries, so : \rho_i ' \simeq \rho_j ' \quad \mbox \quad \sigma_i ' \simeq \sigma_j ' \quad \mbox \quad i,j \;. This proves the theorem.


See also

*
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 ea ...
*
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 ' ...


References

* B. Blackadar, ''Operator Algebras'', Springer, 2006. {{DEFAULTSORT:Schroder-Bernstein theorems for operator algebras C*-algebras Theorems in functional analysis Operator theory Von Neumann algebras