Commutative Radical Banach Algebras

在这个项目中要进行的数学研究的一个相当具体的设置是幂级数，它可以被认为是无限次的多项式。对这种级数的系数的增长（或衰减）施加适当的约束，可以得到幂级数的巴拿赫代数。在代数中计算所有级数的唯一方法可能是取常数项。在这种情况下，我们有一个（交换的）根巴拿赫代数，一个非常奇特的对象。最近，首席研究员已经证明了一个重要的结果，关于这些巴拿赫代数的导数（初等微积分中导数的类似物），从而解决了一个已经开放了三十多年的问题。他开发的技术将被扩展并用于探索其他结构问题。

One fairly concrete setting for the mathematical research to be undertaken in this project is that of power series, which may be thought of as polynomials of infinite degree. Placing an appropriate constraint on the growth (or decay) of the coefficients of such a series gives rise to a Banach algebra of power series. It may happen that the only way to evaluate all of the series in the algebra is by taking the constant term. In this case, one has a (commutative) radical Banach algebra, a very exotic object. Recently, the principal investigator has proved an important result concerning derivations (analogues of the derivative in elementary calculus) of Banach algebras like these, thereby solving a problem that had been open for over thirty years. The techniques he developed will be extended and used to explore other structural questions.