Fundamental Decomposition in Finite von Neumann Algebras

有限冯诺依曼代数的基本分解

基本信息

  • 批准号:
    1800335
  • 负责人:
  • 金额:
    $ 18万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2018
  • 资助国家:
    美国
  • 起止时间:
    2018-08-01 至 2021-07-31
  • 项目状态:
    已结题

项目摘要

The study of operators on Hilbert space became important with the advent of Quantum Mechanics, but in addition, understanding of these operators has proven to be vital to progress in many areas of mathematics. Historically, a method of studying and understanding such operators is to break them down into simpler components, based on spectral decomposition. This consists of describing parts of the operator that behave like multiplication by certain numbers, and to explain how these parts assemble into the whole. One major goal of this project is to advance such understanding of large classes of operators. Another major goal is to study families of operators that arise in various quantum mechanical models, in light of certain deep mathematical conjectures regarding finite dimensional approximations of infinite dimensional objects.More specifically, the principal investigator, together with collaborators, has made advances in recent years on spectral decomposition results for non-selfadjoint elements of finite von Neumann algebras. These are centered around upper triangular forms, analogous to the classical results of Issai Schur for matrices, and both utilize and extend results about hyperinvariant subspaces found recently by Haagerup and Schultz. Particular proposed projects include (a) studying norm convergence properties of bounded operators and (b) extending spectral distribution and upper-triangular form results to unbounded affiliated operators. In related directions, the principal investigator will work on the hyperinvariant subspace problem for elements of tracial von Neumann algebras and to investigate the Murray--von Neumann puzzle, which is akin to the Heisenberg relations. A second area of proposed research concerns the notion of bi-freeness and bi-free products. A third area of proposed research concerns quantum correlations. Recent results of the principal investigator showing non-closedness of the set of quantum correlations for five inputs and two outputs open the door to new understanding of these small cases, which the principal investigator proposes to pursue.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
随着量子力学的出现,希尔伯特空间上算子的研究变得很重要,但除此之外,对这些算子的理解已被证明对于许多数学领域的进步至关重要。 从历史上看,研究和理解此类算子的一种方法是基于谱分解将它们分解为更简单的组件。 这包括描述运算符的某些部分,其行为类似于乘以某些数字,并解释这些部分如何组装成整体。 该项目的一个主要目标是增进对大类操作员的了解。 另一个主要目标是根据关于无限维对象的有限维近似的某些深层数学猜想,研究各种量子力学模型中出现的算子族。更具体地说,主要研究者与合作者近年来在有限冯诺依曼代数的非自共轭元素的谱分解结果方面取得了进展。这些以上三角形式为中心,类似于 Issai Schur 矩阵的经典结果,并且都利用和扩展了 Haagerup 和 Schultz 最近发现的超不变子空间的结果。 具体提议的项目包括(a)研究有界算子的范数收敛特性,以及(b)将谱分布和上三角形式结果扩展到无界附属算子。在相关方向上,首席研究员将致力于追踪冯诺依曼代数元素的超不变子空间问题,并研究类似于海森堡关系的穆雷-冯诺依曼难题。 拟议研究的第二个领域涉及双游离性和双游离产品的概念。 拟议研究的第三个领域涉及量子相关性。 首席研究员最近的结果显示,五个输入和两个输出的量子相关性集的非封闭性,为首席研究员提议追求的这些小案例的新理解打开了大门。该奖项反映了 NSF 的法定使命,并通过使用基金会的智力价值和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Angles between Haagerup–Schultz projections and spectrality of operators
Haagerup-Schultz 投影与算子谱之间的角度
  • DOI:
    10.1016/j.jfa.2021.109027
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    1.7
  • 作者:
    Dykema, Ken;Krishnaswamy-Usha, Amudhan
  • 通讯作者:
    Krishnaswamy-Usha, Amudhan
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Kenneth Dykema其他文献

Some Results in the Hyperinvariant Subspace Problem and Free Probability
超不变子空间问题和自由概率的一些结果
  • DOI:
  • 发表时间:
    2010
  • 期刊:
  • 影响因子:
    0
  • 作者:
    G. H. T. Scuadroni;Ronald Douglas;Scott Miller;Roger Smith;G. Tucci;Kenneth Dykema;Valentina Vega Veglio
  • 通讯作者:
    Valentina Vega Veglio

Kenneth Dykema的其他文献

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{{ truncateString('Kenneth Dykema', 18)}}的其他基金

Great Plains Operator Theory Symposium 2019
2019年大平原算子理论研讨会
  • 批准号:
    1900745
  • 财政年份:
    2019
  • 资助金额:
    $ 18万
  • 项目类别:
    Standard Grant
New Developments in Free Probability and Applications
自由概率及其应用的新进展
  • 批准号:
    1900856
  • 财政年份:
    2019
  • 资助金额:
    $ 18万
  • 项目类别:
    Standard Grant
Research in finite von Neumann algebras
有限冯诺依曼代数研究
  • 批准号:
    1202660
  • 财政年份:
    2012
  • 资助金额:
    $ 18万
  • 项目类别:
    Continuing Grant
Seventh East Coast Operator Algebras Symposium; Fall 2009, College Station, TX
第七届东海岸算子代数研讨会;
  • 批准号:
    0855328
  • 财政年份:
    2009
  • 资助金额:
    $ 18万
  • 项目类别:
    Standard Grant
Sums of Hermitian Operators and Connections to Connes' Embedding Problem; Hyperinvariant Subspaces
厄米算子之和以及与 Connes 嵌入问题的联系;
  • 批准号:
    0901220
  • 财政年份:
    2009
  • 资助金额:
    $ 18万
  • 项目类别:
    Continuing Grant
Functions of operators on Hilbert spaces
希尔伯特空间上的算子函数
  • 批准号:
    0900870
  • 财政年份:
    2009
  • 资助金额:
    $ 18万
  • 项目类别:
    Standard Grant
Free Probability Theory and Applications to Free Group Factors
自由概率论及其在自由群因子中的应用
  • 批准号:
    0600814
  • 财政年份:
    2006
  • 资助金额:
    $ 18万
  • 项目类别:
    Standard Grant
Invariant Subspaces and Free Probability in the Context of von Neumann algebras
冯诺依曼代数背景下的不变子空间和自由概率
  • 批准号:
    0300336
  • 财政年份:
    2003
  • 资助金额:
    $ 18万
  • 项目类别:
    Continuing Grant
Free Probability and Problems in Operator Algebras
算子代数中的自由概率和问题
  • 批准号:
    0070558
  • 财政年份:
    2000
  • 资助金额:
    $ 18万
  • 项目类别:
    Continuing Grant
Mathematical Sciences:Postdoctoral Research Fellowship
数学科学:博士后研究奖学金
  • 批准号:
    9306072
  • 财政年份:
    1993
  • 资助金额:
    $ 18万
  • 项目类别:
    Fellowship Award

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