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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 Stinespring factorization theorem ： ウィキペディア英語版 Stinespring factorization theorem In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C *-algebra as a composition of two completely positive maps each of which has a special form: #A *-representation of ''A'' on some auxiliary Hilbert space ''K'' followed by #An operator map of the form ''T'' → ''VTV'' *. Moreover, Stinespring's theorem is a structure theorem from a C *-algebra into the algebra of bounded operators on a Hilbert space. Completely positive maps are shown to be simple modifications of *-representations, or sometimes called *-homomorphisms. == Formulation == In the case of a unital C *-algebra, the result is as follows: :Theorem. Let ''A'' be a unital C *-algebra, ''H'' be a Hilbert space, and ''B(H)'' be the bounded operators on ''H''. For every completely positive ::$\backslash Phi\; :\; A\; \backslash to\; B(H),$ :there exists a Hilbert space ''K'' and a unital *-homomorphism ::$\backslash pi\; :\; A\; \backslash to\; B(K)$ :such that ::$\backslash Phi(a)\; =\; V^\backslash ast\; \backslash pi\; (a)\; V,$ :where $V:\; H\; \backslash to\; K$ is a bounded operator. Furthermore, we have ::$\backslash |\; \backslash Phi(1)\; \backslash |\; =\; \backslash |\; V\; \backslash |^2.$ Informally, one can say that every completely positive map $\backslash Phi$ can be "lifted" up to a map of the form $V\; (\backslash cdot)\; V^$ *. The converse of the theorem is true trivially. So Stinespring's result classifies completely positive maps. 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「Stinespring factorization theorem」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.