Advances in Abstract Harmonic Analysis
抽象谐波分析的进展
基本信息
- 批准号:RGPIN-2020-06505
- 负责人:
- 金额:$ 1.75万
- 依托单位:
- 依托单位国家:加拿大
- 项目类别:Discovery Grants Program - Individual
- 财政年份:2021
- 资助国家:加拿大
- 起止时间:2021-01-01 至 2022-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The young theory of locally compact (LC) quantum groups, presented in 2000 by Kustermans-Vaes, extends classical Pontryagin duality to a category comprising both LC groups and important deformation algebras arising in mathematical physics. My recent work has contributed to the current rapid development of harmonic analysis on LC quantum groups - giving rise in turn to numerous intriguing novel research projects. The principal aim of my proposal is thus to explore new frontiers of this vast program: this will reveal both novel mathematical objects and approaches to open problems, through links with non-commutative ergodic theory and quantum information, as well as topological centres and Arens (ir)regularity. Our starting point is the following fundamental problem. While the elements of an abstract C*-algebra can be seen as bounded operators on a Hilbert space, there is no such representation for the central objects of abstract harmonic analysis, e.g., the group algebra and its multiplier algebra, the measure algebra. It is thus of great significance to construct a representation model in this setting - and, more generally, for the corresponding algebras over LC quantum groups. My research collaborators and I have made important advances: we have developed a representation model for the completely bounded (right) multipliers of the quantum group algebra, for any LC quantum group. The multiplier algebra can be described via a natural action of the convolution algebra of trace class operators associated with the quantum group, as introduced and studied as well by us. The latter is a fascinating object in its own right, e.g., in relation to the major open problem of the duality between quantum group amenability and co-amenability. As we have shown, the space of trace class operators admits two 'dual' products - quantum versions of convolution and pointwise product - linked by a formula that can be viewed as a tensorial anti-commutation relation. This may form the starting point to develop a duality theory beyond quantum groups. Moreover, we have used our representation to build a functor from LC quantum groups to LC groups that preserves, e.g., compactness and discreteness. Our construction leads to new invariants for LC quantum groups, generalizing Heisenberg's bicharacters, whose explicit calculation is an important task, and which may, similar in spirit to K-theory, form a step towards a classification of LC quantum groups: a program of great potential impact. Our representation also extends to large classes of bidual algebras, establishing important links to topological centre problems, and to the famous Kadison-Singer Problem, which has only been solved in 2015. We plan to tackle the topological centre problem for the Fourier-Stieltjes algebra via my factorization method. Related projects concern topological centres for group actions, and a tensor product version of topological centres.
Kustermans-Vaes在2000年提出的年轻的局部紧量子群理论,将经典的庞特里亚金对偶扩展到一个包含LC群和数学物理中出现的重要变形代数的范畴。我最近的工作有助于LC量子群谐波分析的快速发展,进而产生了许多有趣的新研究项目。因此,我的建议的主要目的是探索这个庞大计划的新领域:这将揭示新的数学对象和开放问题的方法,通过与非交换遍历理论和量子信息的联系,以及拓扑中心和Arens(ir)正则性。我们的出发点是以下基本问题。虽然抽象C*-代数的元素可以被看作是希尔伯特空间上的有界算子,但抽象调和分析的中心对象没有这样的表示,例如,群代数及其乘子代数,测度代数因此,在这种情况下构造一个表示模型是非常有意义的,更一般地说,对于LC量子群上的相应代数也是如此。我和我的研究合作者取得了重要的进展:我们已经为量子群代数的完全有界(右)乘子开发了一个表示模型,适用于任何LC量子群。乘子代数可以通过与量子群相关联的迹类算子的卷积代数的自然作用来描述,正如我们所介绍和研究的那样。后者本身就是一个迷人的对象,例如,关于量子群顺从性和协顺从性之间的对偶性的主要公开问题。正如我们已经证明的,迹类算子空间允许两个“对偶”乘积--卷积和逐点乘积的量子版本--由一个可以被视为张量反对易关系的公式连接。这可能是发展超越量子群的对偶理论的起点。此外,我们已经使用我们的表示来建立从LC量子群到LC群的函子,其保持例如,紧凑性和离散性。我们的建设导致新的LC量子群的不变量,推广海森堡的bicharacters,其明确的计算是一项重要的任务,并可能,类似的精神K-理论,形成一个步骤的分类LC量子群:一个程序的巨大的潜在影响。我们的表示还扩展到大类的双对偶代数,建立了重要的联系,拓扑中心问题,并著名的凯迪森-辛格问题,这是在2015年才解决。我们计划通过我的因式分解方法来解决Fourier-Stieltjes代数的拓扑中心问题。相关的项目涉及群体作用的拓扑中心,以及拓扑中心的张量积版本。
项目成果
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Neufang, Matthias其他文献
Neufang, Matthias的其他文献
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{{ truncateString('Neufang, Matthias', 18)}}的其他基金
Advances in Abstract Harmonic Analysis
抽象谐波分析的进展
- 批准号:
RGPIN-2020-06505 - 财政年份:2022
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual
Advances in Abstract Harmonic Analysis
抽象谐波分析的进展
- 批准号:
RGPIN-2020-06505 - 财政年份:2020
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual
Abstract Harmonic Analysis: New Frontiers
抽象谐波分析:新领域
- 批准号:
RGPIN-2014-06356 - 财政年份:2019
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual
Abstract Harmonic Analysis: New Frontiers
抽象谐波分析:新领域
- 批准号:
RGPIN-2014-06356 - 财政年份:2017
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual
Abstract Harmonic Analysis: New Frontiers
抽象谐波分析:新领域
- 批准号:
RGPIN-2014-06356 - 财政年份:2016
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual
Abstract Harmonic Analysis: New Frontiers
抽象谐波分析:新领域
- 批准号:
RGPIN-2014-06356 - 财政年份:2015
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual
Abstract Harmonic Analysis: New Frontiers
抽象谐波分析:新领域
- 批准号:
RGPIN-2014-06356 - 财政年份:2014
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual
Abstract harmonic analysis - beyond the classical realm
抽象调和分析——超越经典领域
- 批准号:
261894-2008 - 财政年份:2012
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual
Abstract harmonic analysis - beyond the classical realm
抽象调和分析——超越经典领域
- 批准号:
261894-2008 - 财政年份:2011
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual
Abstract harmonic analysis - beyond the classical realm
抽象调和分析——超越经典领域
- 批准号:
261894-2008 - 财政年份:2010
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual
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- 批准号:
RGPIN-2020-06505 - 财政年份:2020
- 资助金额:
$ 1.75万 - 项目类别:
Discovery Grants Program - Individual