Mathematics Sciences: Applications of Complete Boundedness to Operator Algebras

在过去的几年里，算子代数的理论，无论是自伴随的还是非自伴随的，都得到了快速的发展。许多研究人员的最新进展为该理论的自伴随和非自伴随方面提供了进一步的联系。本文主要研究了算子代数的张量积理论、算子代数的完全有界性、上同调理论及其与代数拓扑的联系。还将研究舒尔乘子理论、非自伴代数和自伴代数的导数和自同构的结构以及半有限代数中的紧致理想摄动。算子理论是关于有限矩阵的无限推广。当在运算符集合上定义运算符乘法和加法时，所得到的集合称为运算符代数。上同调是一种理论，研究局部碎片何时可以组合在一起形成一个整体对象。本课题研究算子代数上同调问题。目标是对一个古老的猜想做出贡献，该猜想断言算子代数的某些上同调是平凡的。这意味着局部对象总是可以被拼凑成一个全局对象。＊＊＊

9401540 Smith During the past few years the theory of operator algebras, both selfadjoint and non-selfadjoint has evolved at a rapid pace. Recent developments by many researchers are providing further ties between the selfadjoint and non-selfadjoint aspects of the theory. The proposed work focuses on the theory of tensor products of operator algebras, complete boundedness for operator algebras, cohomology theory, and its connection with algebraic topology. Schur multiplier theory, the structure of derivations and automorphisms of non-selfadjoint and selfadjoint algebras and compact ideal perturbations in semifinite algebras will also be studied. Operator theory is concerned with infinite generalizations of finite matrices. When operator multiplication and addition are defined on a collection of operators the resulting collection is called an operator algebra. Cohomology is a theory that examines when local pieces can be put together to form a global object. This project is concerned with cohomology on operator algebras. The goal is to contribute to an old conjecture which asserts that certain cohomologies of operator algebras are trivial. This would mean that local objects can always be pieced together into a global object. ***