Properties of Banach algebras and their extensions

巴拿赫代数可以有许多不同的性质。两个对比的例子是在封闭单位圆盘上的所有连续复值函数的代数，以及由在封闭圆盘上连续而在圆盘内部解析的函数组成的这个代数的子代数。在第二个代数中，任何在非空开集上为零的函数必须一直为零。这在更大的代数中是完全不同的：事实上，Urysohn引理表明，对于闭盘的任意两个不相交的闭子集，在盘上定义了一个连续的复值函数，在一个闭集上恒定为0，在另一个闭集上恒定为1（这种类型的代数称为正则代数）。大多数可交换巴拿赫代数与这两个代数中的一个或另一个有一些共同的特征。该项目的目的是研究Banach代数，特别是Banach函数代数的各种条件（包括正则性条件），将这些条件相互联系起来，并将这些条件与Banach代数可能满足的其他重要条件联系起来，并研究当你形成各种类型的代数扩展（特别是“代数”扩展，如Arens-Hoffman或Cole扩展）时这些条件的保存或引入。

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).