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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 Finite-dimensional von Neumann algebra ： ウィキペディア英語版 Finite-dimensional von Neumann algebra In mathematics, von Neumann algebras are self-adjoint operator algebras that are closed under a chosen operator topology. When the underlying Hilbert space is finite-dimensional, the von Neumann algebra is said to be a finite-dimensional von Neumann algebra. The finite-dimensional case differs from the general von Neumann algebras in that topology plays no role and they can be characterized using Wedderburn's theory of semisimple algebras. ==Details== Let C''n'' × ''n'' be the ''n'' × ''n'' matrices with complex entries. A von Neumann algebra M is a self adjoint subalgebra in C''n'' × ''n'' such that M contains the identity operator ''I'' in C''n'' × ''n''. Every such M as defined above is a semisimple algebra, i.e. it contains no nilpotent ideals. Suppose ''M'' ≠ 0 lies in a nilpotent ideal of M. Since ''M *'' ∈ M by assumption, we have ''M *M'', a positive semidefinite matrix, lies in that nilpotent ideal. This implies (''M *M'')''k'' = 0 for some ''k''. So ''M *M'' = 0, i.e. ''M'' = 0. The center of a von Neumann algebra M will be denoted by ''Z''(M). Since M is self-adjoint, ''Z''(M) is itself a (commutative) von Neumann algebra. A von Neumann algebra N is called a factor if ''Z''(N) is one-dimensional, that is, ''Z''(N) consists of multiples of the identity ''I''. Theorem Every finite-dimensional von Neumann algebra M is a direct sum of ''m'' factors, where ''m'' is the dimension of ''Z''(M). Proof: By Wedderburn's theory of semisimple algebras, ''Z''(M) contains a finite orthogonal set of idempotents (projections) such that ''PiPj'' = 0 for ''i'' ≠ ''j'', Σ ''Pi'' = ''I'', and :$$ Z(\mathbf M) = \oplus _i Z(\mathbf M) P_i where each ''Z''(M'')Pi'' is a commutative simple algebra. Every complex simple algebras is isomorphic to the full matrix algebra C''k'' × ''k'' for some ''k''. But ''Z''(M'')Pi'' is commutative, therefore one-dimensional. The projections ''Pi'' "diagonalizes" M in a natural way. For ''M'' ∈ M, ''M'' can be uniquely decomposed into ''M'' = Σ ''MPi''. Therefore, :$=\; \backslash oplus\_i\; P\_i\; .$ One can see that ''Z''(M''Pi'') = ''Z''(M'')Pi''. So ''Z''(M''Pi'') is one-dimensional and each M''Pi'' is a factor. This proves the claim. For general von Neumann algebras, the direct sum is replaced by the direct integral. The above is a special case of the central decomposition of von Neumann algebras. 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「Finite-dimensional von Neumann algebra」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.