Mathematical Sciences: Operator Algebras and Operator Theory

拉尔森教授的数学研究项目， 史密斯关注代数的运营商在希尔伯特空间，也对 更一般的Banach空间这项工作的一部分将侧重于 来自子空间格的算子代数或 预测，特别是关于何时两个这样的问题， 通过可逆变换共轭。的财产 也将研究这种代数的非对称性。后者 考虑与单算子理论相联系。 算子代数和间映射的结构性质 他们将被调查。这里特别有意思的是 与矩阵张量有关的完全有界性 任意高维的代数。 这个项目的核心运营商可能是 被认为是一种丰富的数字。他们遵守相同的 算术定律作为数字，但有两个明显的例外，即 乘法的结果取决于 因子被取，并不是每个非零运算符都有一个 相反。数学分析只知道两个数字系统， 真实的和复杂的，但有一个巨大的各种运算符 代数不适合用数字表示的情况可以 通常用算子代数来表示， 后者具有更大程度的灵活性。 代价是理解了结构， 算子代数的分类需要相当大的努力， 比如这个项目要做的工作。

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.