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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 Non-commutative conditional expectation ： ウィキペディア英語版 Non-commutative conditional expectation In mathematics, non-commutative conditional expectation is a generalization of the notion of conditional expectation in classical probability. The space of measurable functions on a $\backslash sigma$-finite measure space $(X,\; \backslash mu)$ is the canonical example of a commutative von Neumann algebra. For this reason, the theory of von Neumann algebras is sometimes referred to as noncommutative measure theory. The intimate connections of probability theory with measure theory suggest that one may be able to extend the classical ideas in probability to a noncommutative setting by studying those ideas on general von Neumann algebras. For von Neumann algebras with a faithful normal tracial state, for example finite von Neumann algebras, the notion of conditional expectation is especially useful. ==Formal definition== A positive, linear mapping $\backslash Phi$ of a von Neumann algebra $\backslash mathcal$ onto a von Neumann algebra $\backslash mathcal$ ($\backslash mathcal$ and $\backslash mathcal$ may be general C *-algebras as well) is said to be a ''conditional expectation'' (of $\backslash mathcal$ onto $\backslash mathcal$) when $\backslash Phi(I)=I$ and $\backslash Phi(R\_1SR\_2)\; =\; R\_1\backslash Phi(S)R\_2$ if $R\_1,\; R\_2\; \backslash in\; \backslash mathcal$ and $S\; \backslash in\; \backslash mathcal$. 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「Non-commutative conditional expectation」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.