Operator Theory and Operator Algebras

算子理论和算子代数

• 批准号: RGPIN-2020-03984
• 负责人: Marcoux, Laurent
• 金额: $ 1.53万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=764414
• 关键词: Operator; Theory; Algebras

Algebras exist as an abstract object in mathematics, and one tries to understand their underlying structure and behaviour. A tool that has often proven to be great value in studying abstract algebras is to study their representations as "concrete" algebras of linear transformations on a particular class of vector spaces known as Hilbert spaces. The goal of the proposed research is to explore diverse open questions in operator theory and operator algebras.  We mention three explicit objectives below. First, we seek to determine whether every continuous representation of a total reduction algebra is completely bounded.  An operator algebra is said to be a total reduction algebra if,  whenever it is represented as an algebra of operators on a Hilbert space, each invariant subspace admits a topological complement which is also invariant for the algebra. This generalises the important notions of amenability and nuclearity of certain algebras of operators.  In previous joint work, we have demonstrated that each commutative total reduction algebra is similar to a nuclear C*-algebra, resolving (in the commutative setting) a problem which was open for over thirty years. Our first problem is a logical next step in the study of non-commutative total reduction algebras. Second, we propose problems from the theory of single linear operators acting on a Hilbert space, transposed to the setting of elements of C*-subalgebras of operators. I am suggesting two new questions (amongst several such problems that I have) of this kind. One of these is the problem of extending a theorem of Specht which describes unitary equivalence of matrices in terms of a comparison of their traces on all words in two non-commuting variables to the setting of C*-algebras. This is joint work with Y. Zhang. We also seek to extend recent results by myself and other co-authors that try to determine the structure of an operator by considering all of its "off-diagonal corners" to the C*-algebra setting. Third, we explore whether every quasidiagonal operator can be approximated arbitrarily well by algebraic, quasidiagonal operators.  Block-diagonal operators are those whose "building blocks" are finite-dimensional matrices, and quasidiagonal operators are those which can be approximated by block-diagonal operators. Quasidiagonality has played a crucial role in the classification of nuclear C*-algebras, and it has proven to be an interesting but extremely subtle property. We have recently obtained a new characterisation of those operators that can be approximated by algebraic, quasidiagonal operators, and we hope that this may lead to an answer to the above question. We also seek to understand when the direct sum of a contractive operator and a normal operator whose spectrum is the unit disc is quasidiagonal. Each of these questions opens several potential avenues of investigation and training for HQP, introducing them to central areas of research in operator theory.

代数是数学中的抽象对象，并且试图了解其潜在的结构和行为。一种经常被证明在研究抽象代数方面具有巨大价值的工具是将其表示为特定类别的矢量空间中的线性变换代数的“混凝土”代数，称为希尔伯特空间。拟议的研究的目的是探索潜水员在操作者理论和操作员代数中的开放问题。我们在下面提到三个明确的目标。首先，我们试图确定总还原代数的每个连续表示是否完全有限。如果操作员代数是总体减少代数，则如果以希尔伯特空间的代数为代数时，每个不变的子空间都会承认拓扑完成，这也是代数的不变性。这一概括是操作员某些代数的合理性和核性的重要说明。在先前的联合工作中，我们已经证明，每个交换性总还原代数类似于核C* - 代数，解决（在交换性的环境中）一个问题已有30多年。我们的第一个问题是研究非交通性总还原代数的逻辑下一步。其次，我们提出了从作用于希尔伯特空间的单个线性操作员理论的问题，转化为运算符c*-subalgebras元素的设置。我建议这样的两个新问题（在我遇到的几个问题中）。其中之一是扩展SpecCht定理的问题，该定理描述了物品的单一等效性，以将两个非交换变量中的所有单词的痕迹比较与C*-Algebras的设置进行比较。这是与Y. Zhang的联合合作。我们还寻求将自己和其他合着者的最新结果扩展到试图通过将其所有“偏角角”考虑到C*-Algebra设置来确定操作员的结构。第三，我们探索是否可以通过代数的准二方运算符将每个准二元操作员任意地近似。区块二角运算符是那些“构建块”是有限维属物品的操作员，而准二方操作员是可以由块 - 划线运算符近似的算子。 quasidiagonation在核C*代数的分类中发挥了至关重要的作用，事实证明，它是一个有趣但非常微妙的特性。最近，我们获得了那些运营商的新特征，这些操作员可以由代数，准二方操作员近似，我们希望这可能会导致上述问题的答案。我们还试图了解何时直接的承包操作员和频谱为单位光盘的正常操作员是quasidiagonal。这些问题中的每一个都为HQP开辟了几种潜在的投资和培训途径，将其介绍给操作员理论研究的中心领域。