Operator Theory and Operator Algebras

代数作为数学中的抽象对象而存在，人们试图理解它们的基本结构和行为。一个经常被证明在研究抽象代数方面很有价值的工具是研究它们在一类特定的被称为希尔伯特空间的向量空间上的线性变换的“具体”代数表示。拟议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，我们确定了全约简代数的每个连续表示是否完全有界，如果一个算子代数被表示为Hilbert空间上的一个算子代数，则每个不变子空间都有一个拓扑补，这个拓扑补对该代数也是不变的。这推广了某些算子代数的顺从性和核性的重要概念。在以前的联合工作中，我们已经证明了每个交换全约简代数类似于核C*-代数，解决了(在交换背景下)一个悬而未决了30多年的问题。我们的第一个问题是研究非交换全约简代数的合乎逻辑的下一步。其次，我们提出了作用在Hilbert空间上的单线性算子理论中的问题，转置到算子的C*-子代数的元素的设置上。我提出了两个新的问题(在我有的几个这样的问题中)。其中一个问题是将Speht的一个定理推广到C*-代数上，该定理通过比较矩阵在两个非对易变量中所有字上的迹来刻画矩阵的酉等价性。这是与张勇合作的作品。我们还试图推广我和其他合著者最近的结果，这些结果试图通过考虑算子到C*-代数设置的所有“非对角角”来确定算子的结构。第三，我们探讨了是否每个拟对角算子都能被代数拟对角算子任意好地逼近，其中块对角算子是那些“积木”是有限维矩阵的算子，而拟对角算子是那些可以被块对角算子逼近的算子。拟对角性在核C*-代数的分类中起着至关重要的作用，它已被证明是一个有趣但极其微妙的性质。我们最近得到了这些算子的一个新的刻画，这些算子可以用代数、拟对角线算子来逼近，我们希望这可能会导致上述问题的答案。我们还试图理解压缩算子和谱为单位圆盘的正规算子的直和何时是拟对角的。这些问题中的每一个都为HQP打开了几个潜在的调查和培训途径，向他们介绍了操作员理论研究的中心领域。

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.