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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 Cuntz algebra ： ウィキペディア英語版 Cuntz algebra In mathematics, the Cuntz algebra $\backslash mathcal\_n$ (after Joachim Cuntz) is the universal C *-algebra generated by ''n'' isometries satisfying certain relations. It is the first concrete example of a separable infinite simple C *-algebra. Every simple infinite C *-algebra contains, for any given ''n'', a subalgebra that has $\backslash mathcal\_n$ as quotient. == Definition and basic properties == Let ''n'' ≥ 2 and ''H'' be a separable Hilbert space. Consider the C *-algebra $\backslash mathcal$ generated by a set ::$\backslash \_^n$ of isometries acting on ''H'' satisfying ::$\backslash sum\_^n\; S\_i\; S\_i^$ * = I. and ::$S\_i^$ * S_j = \delta_ I. Theorem. The concrete C *-algebra $\backslash mathcal$ is isomorphic to the universal C *-algebra $\backslash mathcal$ generated by ''n'' generators ''s''1... ''s''''n'' subject to relations ''si *si'' = 1 for all ''i'' and ∑ ''sisi'' * = 1. The proof of the theorem hinges on the following fact: any C *-algebra generated by ''n'' isometries ''s''1... ''s''''n'' with orthogonal ranges contains a copy of the UHF algebra $\backslash mathcal$ type ''n''∞. Namely $\backslash mathcal$ is spanned by words of the form :$s\_\backslash cdots\; s\_s\_^$ * \cdots s_^ *, k \geq 0. The *-subalgebra $\backslash mathcal$, being approximately finite-dimensional, has a unique C *-norm. The subalgebra $\backslash mathcal$ plays role of the space of ''Fourier coefficients'' for elements of the algebra. A key technical lemma, due to Cuntz, is that an element in the algebra is zero if and only if all its Fourier coefficients vanish. Using this, one can show that the quotient map from $\backslash mathcal$ to $\backslash mathcal$ is injective, which proves the theorem. This universal C *-algebra is called the Cuntz algebra, denoted by $\backslash mathcal\_n$. A C *-algebra is said to be purely infinite if every hereditary C *-subalgebra of it is infinite. $\backslash mathcal\_n$ is a separable, simple, purely infinite C *-algebra. Any simple infinite C *-algebra contains a subalgebra that has $\backslash mathcal\_n$ as a quotient. The UHF algebra $\backslash mathcal$ has a non-unital subalgebra $\backslash mathcal\text{'}$ that is canonically isomorphic to $\backslash mathcal$ itself: In the M''n'' stage of the direct system defining $\backslash mathcal$, consider the rank-1 projection ''e''11, the matrix that is 1 in the upper left corner and zero elsewhere. Propagate this projection through the direct system. At the M''nk'' stage of the direct system, one has a rank ''n''''k'' - 1 projection. In the direct limit, this gives a projection ''P'' in $\backslash mathcal$. The corner :$P\; \backslash mathcal\; P\; =\; \backslash mathcal$ is isomorphic to $\backslash mathcal$. The *-endomorphism Φ that maps $\backslash mathcal$ onto $\backslash mathcal\text{'}$ is implemented by the isometry ''s''1, i.e. Φ(·) = ''s''1(·)''s''1 *. $\backslash ;\backslash mathcal\_n$is in fact the crossed product of $\backslash mathcal$ with the endomorphism Φ. 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「Cuntz 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.