Operator Theory and Operator Algebras

代数作为数学中的抽象对象存在，人们试图理解它们的基本结构和行为。在研究抽象代数时，一个经常被证明是很有价值的工具是研究它们在一类特殊的向量空间（称为希尔伯特空间）上作为线性变换的“具体”代数的表示。该研究的目的是探索算子理论和算子代数中的各种开放性问题。 我们在下面提到三个明确的目标。首先，我们试图确定是否每一个连续表示的全约简代数是完全有界的。 一个算子代数被称为全约化代数，如果当它被表示为希尔伯特空间上的算子代数时，每个不变子空间都有一个拓扑补，这个拓扑补对于这个代数也是不变的。这概括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化代数都类似于核C*-代数，解决了（在交换设置中）一个开放了三十多年的问题。我们的第一个问题是在非交换全约化代数的研究中合乎逻辑的下一步。其次，我们提出的问题，从理论上的单一线性算子作用在希尔伯特空间，转置设置的C*-子代数的运营商的元素。我提出了两个新的问题（在我的几个这样的问题中）。其中之一是问题的延伸定理的Specht描述酉等价矩阵的比较其痕迹的所有字在两个非交换变量的设置的C*-代数。这是与Y的合作。张某我们还试图扩展最近的结果，我和其他合著者，试图确定一个运营商的结构，考虑其所有的“非对角”的C*-代数设置。第三，我们探讨是否每一个准对角算子可以近似任意好的代数，准对角算子。 块对角算子是那些其“构建块”是有限维矩阵的算子，而准对角算子是那些可以由块对角算子近似的算子。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。我们最近得到了一个新的特征，这些运营商可以近似代数，拟对角运营商，我们希望这可能会导致上述问题的答案。我们还试图了解当压缩算子和正常运营商的频谱是单位圆盘的直和是拟对角。这些问题中的每一个都为HQP的调查和培训开辟了几条潜在的途径，将他们引入了算子理论的中心研究领域。

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.