Properties of Banach algebras and their extensions

Banach代数可以有许多不同的性质。两个对比的例子是代数的所有连续复值函数的封闭单位光盘，和子代数这个代数组成的这些职能是连续的封闭光盘和分析的内部光盘。在第二个代数中，任何在某个非空开集上为零的函数必定是常零的。在更大的代数中，情况并非如此：事实上，Urysohn引理表明，对于闭圆盘的任何两个不相交的闭子集，存在连续的，定义在圆盘上的复值函数，在一个闭集上常为0，在另一个闭集上常为1（这种类型的代数称为正则代数）。大多数交换Banach代数与这两个代数中的一个或另一个有一些共同的特征。该项目的目的是调查各种条件（包括正则性条件）的Banach代数，特别是Banach函数代数，将这些条件相互联系起来，并与其他重要的条件，Banach代数可能满足，并研究在构造代数的各种扩张时，这些条件的保持或引入（特别是“代数”扩展，如Arens-Hoffman或科尔扩展）。

Banach algebras can have many different properties. Two contrasting examples are the algebra of all continuous complex-valued functions on the closed unit disc, and the subalgebra of this algebra consisting of those functions which are continuous on the closed disc and analytic on the interior of the disc. In the second of these algebras, any function which is zero throughout some non-empty open set must be constantly zero. This is very much not the case in the bigger algebra: indeed Urysohn's lemma shows that for any two disjoint closed subsets of the closed disc, there is a continuous, complex-valued function defined on the disc which is constantly 0 on one closed set and constantly 1 on the other (algebras of this type are called regular algebras).Most commutative Banach algebras have some features in common with one or the other of these two algebras. The aim of this project is to investigate a variety of conditions (including regularity conditions) for Banach algebras, especially Banach function algebras, to relate these conditions to each other, and to other important conditions that Banach algebras may satisfy, and to investigate the preservation or introduction of these conditions when you form various types of extension of the algebras (especially 'algebraic' extensions such as Arens-Hoffman or Cole extensions).