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.

代数作为数学中的抽象对象而存在，人们试图理解它们的底层结构和行为，在研究抽象代数时经常被证明具有巨大价值的工具是研究它们作为特定线性变换的“具体”代数的表示。所提出的研究的目标是探索算子理论和算子代数中的各种开放问题，首先，我们试图确定是否每个连续表示都是总约简。代数是完全有界的，如果只要将算子代数表示为希尔伯特空间上的算子代数，每个不变子空间都承认一个对于代数的重要概念也是不变的拓扑补。某些算子代数的顺应性和核性在之前的联合工作中，我们已经证明了每个交换全约简代数都类似于核。 C*-代数，解决（在交换环境中）一个开放了三十多年的问题。我们的第一个问题是非交换全约简代数研究中合乎逻辑的下一步。作用于希尔伯特空间的单个线性算子，转置为算子的 C* 子代数的元素设置。我提出了两个此类新问题（在我遇到的几个此类问题中），其中之一是扩展问题。 Specht 定理，通过将两个非交换变量中所有词的迹与 C* 代数的设置进行比较来描述矩阵的酉等价性。这是我们与 Y.Zhang 的联合工作。我和其他合著者的最新结果试图通过考虑 C* 代数设置的所有“非对角”来确定算子的结构。第三，我们探讨是否每个准对角算子。可以用代数任意近似，拟对角算子的“构建块”是有限维矩阵，而拟对角算子可以用块对角算子来近似。核 C* 代数的分类，它已被证明是一个有趣但极其微妙的属性，我们最近获得了这些算子的新特征，可以通过以下方式近似。代数、拟对角算子，我们希望这可以解答上述问题。我们还试图理解收缩算子和其谱为单位圆盘的正规算子的直和何时是拟对角的。为 HQP 开辟了几种潜在的调查和培训途径，将他们引入算子理论的中心研究领域。