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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 Locally compact quantum group ： ウィキペディア英語版 Locally compact quantum group A locally compact quantum group is a relatively new C *-algebraic approach toward quantum groups that generalizes the Kac-algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicative unitaries have enjoyed some success but have also encountered several technical problems. One of the main features distinguishing this new approach from its predecessors is the axiomatic existence of left and right invariant weights. This gives a noncommutative analogue of left and right Haar measures on a locally compact Hausdorff group. == Definitions == Before we can even begin to properly define a locally compact quantum group, we first need to define a number of preliminary concepts and also state a few theorems. Definition (weight). Let $A$ be a C *-algebra, and let $A\_$ denote the set of positive elements of $A$. A weight on $A$ is a function $\backslash phi:\; A\_\; \backslash to\; ()$ such that * $\backslash phi(a\_\; +\; a\_)\; =\; \backslash phi(a\_)\; +\; \backslash phi(a\_)$ for all $a\_,a\_\; \backslash in\; A\_$, and * $\backslash phi(r\; \backslash cdot\; a)\; =\; r\; \backslash cdot\; \backslash phi(a)$ for all $r\; \backslash in$ a weight on $A$. We say that $\backslash phi$ is a K.M.S. weight ('K.M.S.' stands for 'Kubo-Martin-Schwinger') on $A$ if and only if $\backslash phi$ is a ''proper weight'' on $A$ and there exists a norm-continuous one-parameter group $(\backslash sigma\_)\_\; =\; \backslash phi$ for all $t\; \backslash in\; \backslash mathbb$, and * for every $a\; \backslash in\; \backslash text(\backslash sigma\_)$, we have is a dense subset of $B$), then we can uniquely extend $\backslash pi$ to a *-homomorphism $\backslash overline:\; M(A)\; \backslash to\; M(B)$. Theorem 3. If $\backslash omega:\; A\; \backslash to\; \backslash mathbb$ is a state (i.e., a positive linear functional of norm $1$) on $A$, then we can uniquely extend $\backslash omega$ to a state $\backslash overline:\; M(A)\; \backslash to\; \backslash mathbb$ on $M(A)$. Definition (Locally compact quantum group). A (C *-algebraic) locally compact quantum group is an ordered pair $\backslash mathcal\; =\; (A,\backslash Delta)$, where $A$ is a C *-algebra and $\backslash Delta:\; A\; \backslash to\; M(A\; \backslash otimes\; A)$ is a ''non-degenerate'' *-homomorphism called the co-multiplication, that satisfies the following four conditions: * The co-multiplication is co-associative, i.e., $\backslash overline\; \backslash circ\; \backslash Delta\; =\; \backslash overline\; \backslash circ\; \backslash Delta$. * The sets $\backslash left\backslash ,\; ~\; a\; \backslash in\; A\; \backslash right\backslash \}$ and $\backslash left\backslash (\backslash Delta(a))\; ~\; \backslash big|\; ~\; \backslash omega\; \backslash in\; A^,\; ~\; a\; \backslash in\; A\; \backslash right\backslash \}$ are linearly dense subsets of $A$. * There exists a faithful K.M.S. weight $\backslash phi$ on $A$ that is left-invariant, i.e., $\backslash phi\; \backslash !\; \backslash left(\; \backslash overline(1\_)\; \backslash cdot\; \backslash phi(a)$ for all $\backslash omega\; \backslash in\; A^$ and $a\; \backslash in\; \backslash mathcal\_^$. * There exists a K.M.S. weight $\backslash psi$ on $A$ that is right-invariant, i.e., $\backslash psi\; \backslash !\; \backslash left(\; \backslash overline(\backslash Delta(a))\; \backslash right)\; =\; \backslash overline(1\_)\; \backslash cdot\; \backslash psi(a)$ for all $\backslash omega\; \backslash in\; A^$ and $a\; \backslash in\; \backslash mathcal\_^$. From the definition of a locally compact quantum group, it can be shown that the right-invariant K.M.S. weight $\backslash psi$ is automatically faithful. Therefore, the faithfulness of $\backslash psi$ is a redundant condition and does not need to be postulated. 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「Locally compact quantum group」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.