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| 共轭域上的*代数及其*表示 |
| *-algebras and their *-representations over conjugate fields |
| 投稿时间：2024-12-26 修订日期：2025-03-24 |
| DOI： |
| 中文关键词: 共轭域 *代数 Hermite空间 *表示 酉表示 |
| 英文关键词: conjugate field *-algebra Hermitian space *-representation unitary representation |
| 基金项目:国家自然科学基金青年项目，拓扑动力系统中几类回复性的研究，项目号:12101129 |
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| 中文摘要: |
| 算子代数是泛函分析的一个重要方向,它的基本内容是C*代数与von Neumann代数及其表示理论.本文主要研究共轭域上的*代数及其*表示,为研究更一般的代数提供新的工具和方法.首先通过引入共轭域的概念,将复数域上的共轭运算推广到更一般的域上,证明了非平凡共轭域本质上是某个可分二次扩张,将复数域上的*代数推广为共轭域上的*代数,并探讨了其幺元化的泛性质.进一步,本文还参考了Hilbert空间理论,定义了共轭域上的Hermite空间,并研究了其伴随算子理论,从C*代数在Hilbert空间上的表示理论中得到启发,定义了*代数在Hermite空间上的*表示.最后作为应用,证明了群的酉表示与其群代数的保幺*表示之间存在一一对应关系,为群表示论与代数表示论之间的联系提供了新的视角. |
| 英文摘要: |
| Operator algebra is an important direction of functional analysis. Its basic content is C*-algebra and von Neumann algebra and their representation theory. This paper mainly studies the *-algebra and its *-representation on conjugate fields, providing new tools and methods for studying more general algebras. Firstly, by introducing the concept of conjugate fields, the conjugate operation on complex fields is generalized to more general fields. It is proved that non-trivial conjugate fields are essentially some separable quadratic extensions. The *-algebra on complex fields is generalized to the *-algebra on conjugate fields, and its universal properties of unitization are discussed. Furthermore, this paper also refers to the Hilbert space theory, defines the Hermite space on conjugate fields, and studies its adjoint operator theory. Inspired by the representation theory of C*-algebra on Hilbert space, the *-representation of *-algebra on Hermitian space is defined. Finally, as an application, it is proved that there is a one-to-one correspondence between the unitary representation of a group and the unital *-representation of its group algebra, providing a new perspective for the connection between group representation theory and algebraic representation theory. |
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