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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Schröder–Bernstein theorems for operator algebras ： ウィキペディア英語版 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 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 :$M\; =\; M\_0\; \backslash supset\; N\_0$ 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 :$M\; =\; M\_0\; \backslash supset\; N\_0\; \backslash supset\; M\_1.$ By induction, :$M\; =\; M\_0\; \backslash supset\; N\_0\; \backslash supset\; M\_1\; \backslash supset\; N\_1\; \backslash supset\; M\_2\; \backslash supset\; N\_2\; \backslash supset\; \backslash cdots\; .$ It is clear that :$R\; =\; \backslash cap\_\; M\_i\; =\; \backslash cap\_\; N\_i.$ Let :$M\; \backslash ominus\; N\; \backslash stackrel\; M\; \backslash cap\; (N)^.$ 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 :$M\_i\; \backslash ominus\; N\_i\; \backslash sim\; M\; \backslash ominus\; N\; \backslash quad\; \backslash mbox\; \backslash quad\; i.$ The theorem now follows from the countable additivity of ~. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Schröder–Bernstein theorems for operator algebras」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.