Mathematical Sciences: Operator Algebras and Operator Theory

Larson和Smith教授的数学研究项目涉及Hilbert空间和更一般的Banach空间上的算子代数。这项工作的一部分将集中在从子空间或投影格中得到的算子代数，特别是两个这样的算子代数何时通过可逆变换共轭的问题。还将研究这类代数的相对性。后一种考虑与单算子理论相联系。我们将研究算子代数和它们之间映射的结构性质。这里特别感兴趣的是与任意高维矩阵代数张量有关的概念，如完全有界性。处于这个项目核心的运算符可能被认为是一个丰富的数字物种。它们遵守与数字相同的算术定律，但有两个明显的例外，即乘法的结果取决于因子的取值顺序，并且并不是每个非零运算符都有一个逆运算符。数学分析只知道两个数字系统，实数和复数，但运算符代数种类繁多。不适合用数字表示的情况通常可以用算符代数以某种方式表示，这要归功于后者提供的更大程度的灵活性。这样做的代价是，理解算子代数的结构和分类需要相当大的努力，例如这个项目中要进行的工作。

The mathematical research project of Professors Larson and Smith concerns algebras of operators on Hilbert space and also on more general Banach spaces. Part of this work will focus on operator algebras that come from lattices of subspaces or projections, in particular on the question of when two such are conjugate by an invertible transformation. The property of relexivity for such algebras will also be studied. The latter consideration makes contact with single operator theory. Structural properties of operator algebras and of maps between them will be investigated. Of especial interest here are notions like complete boundedness that relate to tensoring with matrix algebras of arbitrarily high dimension. The operators that lie at the heart of this project may be thought of as a species of enriched numbers. They obey the same laws of arithmetic as numbers with two notable exceptions, namely that the result of multiplication depends on the order in which the factors are taken, and not every nonzero operator has an inverse. Mathematical analysis knows only two number systems, the real and complex, but there is an immense variety of operator algebras. Situations unsuitable for numerical representation can often be represented in some fashion by operator algebras, thanks to the greater degree of flexibility available with the latter. The price for this is that understanding the structure and classification of operator algebras requires considerable effort, as for instance the work to be undertaken in this project.