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| The [[Cantor–Bernstein–Schroeder theorem]], from [[set theory]], has analogs in the context [[operator algebras]]. This article discusses such operator-algebraic results.
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| == For von Neumann algebras ==
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| 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''.
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| For closed subspaces ''M'' and ''N'' where projections ''P<sub>M</sub>'' and ''P<sub>N</sub>'', onto ''M'' and ''N'' respectively, are elements of '''M''', ''M'' « ''N'' if ''P<sub>M</sub>'' « ''P<sub>N</sub>''. | |
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| The '''Schröder–Bernstein theorem''' states that if ''M'' « ''N'' and ''N'' « ''M'', then ''M'' ~ ''N''.
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| 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
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| :<math>M = M_0 \supset N_0</math>
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| where ''N''<sub>0</sub> is an isometric copy of ''N'' in ''M''. By assumption, it is also true that, ''N'', therefore ''N''<sub>0</sub>, contains an isometric copy ''M''<sub>1</sub> of ''M''. Therefore one can write
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| :<math>M = M_0 \supset N_0 \supset M_1.</math>
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| By induction,
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| :<math>M = M_0 \supset N_0 \supset M_1 \supset N_1 \supset M_2 \supset N_2 \supset \cdots .</math>
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| It is clear that
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| :<math>R = \cap_{i \geq 0} M_i = \cap_{i \geq 0} N_i.</math> | |
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| Let
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| :<math>M \ominus N \stackrel{\mathrm{def}}{=} M \cap (N)^{\perp}.</math> | |
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| So
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| :<math>
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| M = \oplus_{i \geq 0} ( M_i \ominus N_i ) \quad \oplus \quad \oplus_{j \geq 0} ( N_j \ominus M_{j+1}) \quad \oplus R
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| </math>
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| and
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| :<math>
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| N_0 = \oplus_{i \geq 1} ( M_i \ominus N_i ) \quad \oplus \quad \oplus_{j \geq 0} ( N_j \ominus M_{j+1}) \quad \oplus R.
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| </math>
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| Notice
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| :<math>M_i \ominus N_i \sim M \ominus N \quad \mbox{for all} \quad i.</math>
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| The theorem now follows from the countable additivity of ~.
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| == Representations of C*-algebras ==
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| There is also an analog of Schröder–Bernstein for representations of [[C*-algebras]]. If ''A'' is a C*-algebra, a '''[[Gelfand Naimark theorem|representation]]''' of ''A'' is a *-homomorphism ''φ'' from ''A'' into ''L''(''H''), the bounded operators on some Hilbert space ''H''.
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| 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''.
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| Two representations ''φ''<sub>1</sub> and ''φ''<sub>2</sub>, on ''H''<sub>1</sub> and ''H''<sub>2</sub> respectively, are said to be '''unitarily equivalent''' if there exists an unitary operator ''U'': ''H''<sub>2</sub> → ''H''<sub>1</sub> such that ''φ''<sub>1</sub>(''a'')''U'' = ''Uφ''<sub>2</sub>(''a''), for every ''a''.
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| In this setting, the '''Schröder–Bernstein theorem''' reads:
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| :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.
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| 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
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| :<math>\rho = \rho_1 \simeq \rho_1 ' \oplus \sigma_1 \quad \mbox{where} \quad \sigma_1 \simeq \sigma.</math>
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| In turn,
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| :<math>\rho_1 \simeq \rho_1 ' \oplus (\sigma_1 ' \oplus \rho_2) \quad \mbox{where} \quad \rho_2 \simeq \rho .</math>
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| By induction,
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| :<math>
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| \rho_1 \simeq \rho_1 ' \oplus \sigma_1 ' \oplus \rho_2' \oplus \sigma_2 ' \cdots \simeq ( \oplus_{i \geq 1} \rho_i ' ) \oplus
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| ( \oplus_{i \geq 1} \sigma_i '),
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| </math>
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| and
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| :<math> | |
| \sigma_1 \simeq \sigma_1 ' \oplus \rho_2' \oplus \sigma_2 ' \cdots \simeq ( \oplus_{i \geq 2} \rho_i ' ) \oplus
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| ( \oplus_{i \geq 1} \sigma_i ').
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| </math>
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| Now each additional summand in the direct sum expression is obtained using one of the two fixed partial isometries, so
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| :<math>
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| \rho_i ' \simeq \rho_j ' \quad \mbox{and} \quad \sigma_i ' \simeq \sigma_j ' \quad \mbox{for all} \quad i,j \;.
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| </math>
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| This proves the theorem.
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| ==See also==
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| * [[Schroeder–Bernstein theorem]] for plain sets
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| * [[Schroeder–Bernstein theorem for measurable spaces]]
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| * [[Schröder–Bernstein theorems for Banach spaces]]
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| * [[Schröder–Bernstein property]]
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| ==References==
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| *B. Blackadar, ''Operator Algebras'', Springer, 2006.
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| {{DEFAULTSORT:Schroder-Bernstein theorems for operator algebras}}
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| [[Category:Von Neumann algebras]]
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| [[Category:C*-algebras]]
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Nice to satisfy you, my name is Numbers Held although I don't really like becoming called like that. I am a meter reader but I strategy on altering it. What I adore performing is playing baseball but I haven't produced a dime with it. For a while she's been in South Dakota.
Also visit my site :: http://www.neweracinema.com