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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 Tomita–Takesaki theory ： ウィキペディア英語版 Tomita–Takesaki theory In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors, and has led to a good structure theory for these previously intractable objects. The theory was introduced by , but his work was hard to follow and mostly unpublished, and little notice was taken of it until wrote an account of Tomita's theory. ==Modular automorphisms of a state== Suppose that ''M'' is a von Neumann algebra acting on a Hilbert space ''H'', and Ω is a separating and cyclic vector of ''H'' of norm 1. (Cyclic means that ''MΩ'' is dense in ''H'', and separating means that the map from ''M'' to ''MΩ'' is injective.) We write φ for the state $\backslash phi(x)=(x\backslash Omega,\backslash Omega)$ of ''M'', so that ''H'' is constructed from φ using the GNS construction. We can define an unbounded antilinear operator ''S''0 on ''H'' with domain ''MΩ'' by setting $S\_0(m\backslash Omega)=m^$ *\Omega for all ''m'' in ''M'', and similarly we can define an unbounded antilinear operator ''F''0 on ''H'' with domain ''M'Ω'' by setting $F\_0(m\backslash Omega)=m^$ *\Omega for ''m'' in ''M''′, where ''M''′ is the commutant of ''M''. These operators are closable, and we denote their closures by ''S'' and ''F'' = ''S'' *. They have polar decompositions $S=J|S|=J\backslash Delta^=\backslash Delta^J$ $F=J|F|=J\backslash Delta^=\backslash Delta^J$ where $J=J^=J^$ is an antilinear isometry called the modular conjugation and $\backslash Delta=S^$ *S=FS is a positive self adjoint operator called the modular operator. The main result of Tomita–Takesaki theory states that: $\backslash Delta^M\backslash Delta^\; =\; M$ for all ''t'' and that $JMJ=M\text{'},$ the commutant of ''M''. :There is a 1-parameter family of modular automorphisms σφ''t'' of ''M'' associated to the state φ, defined by $\backslash sigma^(x)=\backslash Delta^x\backslash Delta^$ 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「Tomita–Takesaki theory」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.