Banach algebra and operator space techniques in topological group theory

拓扑群论中的巴纳赫代数和算子空间技术

基本信息

  • 批准号:
    EP/I002316/1
  • 负责人:
  • 金额:
    $ 2.87万
  • 依托单位:
  • 依托单位国家:
    英国
  • 项目类别:
    Research Grant
  • 财政年份:
    2010
  • 资助国家:
    英国
  • 起止时间:
    2010 至 无数据
  • 项目状态:
    已结题

项目摘要

There are two major strands lying behind our research. One is that of classical `harmonic analysis', in which functions are `separated into their periodic parts'. For example, a sound wave can be split into different frequencies, and once one is working in the `frequency domain' it is possible to process the signal-- for example, frequencies which the human ear cannot process can be removed, one of the ideas underpinning modern digital audio compression. This idea goes back nearly 300 hundred years to Fourier, who stated that `each function can be written as the infinite sum of trigonometric functions'. The developments of this idea lie at the base of much analysis in the subsequent centuries. In due course, in the 20th century, the subject of harmonic analysis became a huge and sophisticated subject; it considers functions and measures on general `locally compact groups' by inspecting the `Fourier transform' of these objects.The second strand behind our subject is that of bounded linear operators on Hilbert spaces; these operators generalize to infinitely many dimensions the idea of a matrix acting on a Euclidean space of finitely many dimensions. Hilbert spaces arise naturally in harmonic analysis: for example, a sound wave has infinitely many frequencies, each frequency giving rise to a dimension in `frequency space'. It is natural to expect that a sensible operation on frequency space will be linear-- the notion of adding sound waves, for example. The theory of operators on a Hilbert space also has numerous applications in the mathematical foundations of Quantum Mechanics, and historically arose in this area. These days this area of mathematics is refered to as the study of `operator algebras', and is a vast area of research with links across pure and applied mathematics.In the last 15--20 years a new link between operator algebras and harmonic analysis has arisen in the theory of `operator spaces'. This uses spaces of bounded linear operators on Hilbert spaces as a model for spaces as well as algebras. The natural maps between such objects are the `completely bounded operators'. Such maps seem to single out an important subclass of the collection of all bounded operators.Now this subject is being applied to modern harmonic analysis; it has already been shown that many results developed separately are really special cases of more general principles that have been discovered recently.It is the purpose of our workshop to bring together about 30 distinguished experts inboth aspects of our subject for a period of intensive study and lectures, intended for both the other experts and for graduate students and colleagues in other areas of mathematics. A number of UK-based people will attend these lectures.We expect existing collaborations between small groups of mathematicians to be consolidated and that new ones will form; these collaborations will generate, both within the workshop itself and in the future, some solutions to the open problems that we know of, and allow the participants to form new syntheses and approaches using ideas that they learn at the workshop. These new results will be disseminated in open access preprint servers and in international research journals.
在我们的研究背后有两条主线。一种是经典的“调和分析”,其中函数被“分成其周期部分”。例如,声波可以被分成不同的频率,一旦人们在“频域”工作,就可以处理信号-例如,人耳无法处理的频率可以被去除,这是现代数字音频压缩的基础思想之一。这个想法可以追溯到近300年前的傅立叶,他说“每个函数都可以写成三角函数的无限和”。这一思想的发展是后来几个世纪许多分析的基础。在适当的时候,在世纪,调和分析的主题成为一个巨大的和复杂的主题;它考虑的功能和措施一般“局部紧群”通过检查“傅立叶变换”这些对象。这些算子将作用在无穷多维欧氏空间上的矩阵的概念推广到无穷多维。希尔伯特空间在谐波分析中自然出现:例如,声波有无限多个频率,每个频率在“频率空间”中产生一个维度。人们很自然地期望频率空间上的合理操作将是线性的-例如,添加声波的概念。希尔伯特空间上的算子理论在量子力学的数学基础中也有许多应用,并且历史上出现在这个领域。这些天这一领域的数学是指作为研究的“运营商代数”,是一个广阔的领域的研究与联系,跨越纯数学和应用mathematics.In过去的15- 20年之间的一个新的联系运营商代数和谐波分析已经出现在理论的“运营商空间”。它使用希尔伯特空间上的有界线性算子空间作为空间和代数的模型。这些对象之间的自然映射是“完全有界算子”。这类映射似乎是所有有界算子集合中的一个重要子类。已经表明,许多单独发展的结果实际上是最近发现的更普遍原理的特例。我们研讨会的目的是召集大约30位在我们的主题的两个方面的杰出专家进行一段时间的密集讨论。研究和讲座,旨在为其他专家和研究生和同事在其他领域的数学。一些英国人将参加这些讲座。我们希望现有的合作小组之间的数学家将得到巩固,新的将形成;这些合作将产生,无论是在工作坊本身和未来,一些解决方案,我们知道的开放问题,并允许参与者形成新的综合和方法,他们在工作坊学习的想法。这些新成果将在开放获取预印本服务器和国际研究期刊上传播。

项目成果

期刊论文数量(6)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Idempotent states on locally compact quantum groups
局部紧量子群上的幂等态
  • DOI:
    10.48550/arxiv.1102.2051
  • 发表时间:
    2011
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Salmi P
  • 通讯作者:
    Salmi P
Subgroups and strictly closed invariant C*-subalgebras
子群和严格闭不变 C* 子代数
  • DOI:
    10.48550/arxiv.1110.5459
  • 发表时间:
    2011
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Salmi P
  • 通讯作者:
    Salmi P
Closed quantum subgroups of locally compact quantum groups
局部紧量子群的闭量子子群
  • DOI:
    10.48550/arxiv.1203.5063
  • 发表时间:
    2012
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Daws M
  • 通讯作者:
    Daws M
Multipliers of locally compact quantum groups via Hilbert C *-modules
通过 Hilbert C * 模的局部紧量子群乘子
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H Dales其他文献

H Dales的其他文献

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{{ truncateString('H Dales', 18)}}的其他基金

Multi-norms and multi-Banach algebras
多范数和多Banach代数
  • 批准号:
    EP/H019405/1
  • 财政年份:
    2009
  • 资助金额:
    $ 2.87万
  • 项目类别:
    Research Grant
Approximate amenability for Banach algebras
Banach 代数的近似适用性
  • 批准号:
    EP/E026664/1
  • 财政年份:
    2007
  • 资助金额:
    $ 2.87万
  • 项目类别:
    Research Grant

相似国自然基金

李代数的权表示
  • 批准号:
    10371120
  • 批准年份:
    2003
  • 资助金额:
    13.0 万元
  • 项目类别:
    面上项目

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CQIS: Operator algebra and Quantum Information Theory
CQIS:算子代数和量子信息论
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Studying generalised Thompson's group with tools from geometric group theory and operator algebra
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顶点代数和 Moonshine 的新方向
  • 批准号:
    22K03264
  • 财政年份:
    2022
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Noncommutative Functions, Algebra and Operator Analysis
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    2155033
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    2022
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Annual Spring Institute on Non-Commutative Geometry and Operator Algebra 2020
2020 年春季非交换几何与算子代数研究所
  • 批准号:
    2000214
  • 财政年份:
    2020
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    $ 2.87万
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East Coast Operator Algebra Symposium (ECOAS) 2020
东海岸算子代数研讨会 (ECOAS) 2020
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    2035183
  • 财政年份:
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Crossed Products of Operator Algebra Dynamical Systems
算子代数动力系统的叉积
  • 批准号:
    504008-2017
  • 财政年份:
    2019
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    $ 2.87万
  • 项目类别:
    Alexander Graham Bell Canada Graduate Scholarships - Doctoral
2019 East Coast Operator Algebra Symposium
2019东海岸算子代数研讨会
  • 批准号:
    1936283
  • 财政年份:
    2019
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    $ 2.87万
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The 2018 East Coast Operator Algebra Symposium
2018东海岸算子代数研讨会
  • 批准号:
    1837227
  • 财政年份:
    2018
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    $ 2.87万
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    Standard Grant
Crossed Products of Operator Algebra Dynamical Systems
算子代数动力系统的叉积
  • 批准号:
    504008-2017
  • 财政年份:
    2018
  • 资助金额:
    $ 2.87万
  • 项目类别:
    Alexander Graham Bell Canada Graduate Scholarships - Doctoral
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