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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 Universal C*-algebra ： ウィキペディア英語版 Universal C*-algebra In mathematics, more specifically in the theory of C *-algebras, a universal C *-algebra is one characterized by a universal property. A universal C *-algebra can be expressed as a presentation, in terms of generators and relations. One requires that the generators must be realizable as bounded operators on a Hilbert space, and that the relations must prescribe a uniform bound on the norm of each generator. For example, the universal C *-algebra generated by a unitary element ''u'' has presentation <''u'' | ''u *u'' = ''uu *'' = 1>. By the functional calculus, this C *-algebra is the continuous functions on the unit circle in the complex plane. Any C *-algebra generated by a unitary element is the homomorphic image of this universal C *-algebra. We next describe a general framework for defining a large class of these algebras. Let ''S'' be a countable semigroup (in which we denote the operation by juxtaposition) with identity ''e'' and with an involution * such that * $e^$ * = e, \quad * $(x^$ *)^ * = x,\quad * $(x\; y)^$ * = y^ * x^ *.\quad Define : ''l''1(''S'') is a Banach space, and becomes an algebra under ''convolution'' defined as follows: :$(\backslash star\; \backslash psi)(x)\; =\; \backslash sum\_$ Theorem. ''l''1(''S'') is a C *-algebra with identity. The universal C *-algebra of contractions generated by ''S'' is the C *-enveloping algebra of ''l''1(''S''). We can describe it as follows: For every state ''f'' of ''l''1(''S''), consider the cyclic representation π''f'' associated to ''f''. Then :$\backslash |\backslash varphi\backslash |\; =\; \backslash sup\_\; \backslash |\backslash pi\_f(\backslash varphi)\backslash |$ is a C *-seminorm on ''l''1(''S''), where the supremum ranges over states ''f'' of ''l''1(''S''). Taking the quotient space of ''l''1(''S'') by the two-sided ideal of elements of norm 0, produces a normed algebra which satisfies the C *-property. Completing with respect to this norm, yields a C *-algebra. ==References== * 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「Universal C*-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.