"Derivations, cohomology groups and second duals of Banach algebras"

我们在高中(或大学)的第一堂微积分课上学到，如果f和g是两个函数，其导数D(F)和D(G)存在，则乘积函数Fg也有导数D(Fg)，它满足D(Fg)=D(F)g+Fd(G)。在高等数学中，有一些由元素组成的结构--通常称为向量--可以相加或相乘，但与函数的乘法不同，这些结构的向量的乘法通常可能无法满足fg=gf。让我们把微分看作是一个映射，它向每个可微函数分配另一个服从微分规则的函数(它的导数D(F))。在抽象结构上存在类似的映射D，其向该结构中的每个向量f分配相同或另一相关结构中的向量D(F)，受数字的加法和乘法的保留，并且以与微分相同的方式作用于两个向量的乘积，即D(Fg)=D(F)g+fd(G)。这些映射称为派生。公式D(F)=fg-gf是最简单的导数之一。这里，g是固定的向量，f在D作用的给定结构的向量中变化。这些派生被称为内部派生。我们建议找到某些结构上的导子公式，并研究那些抽象结构，这些抽象结构具有这样的性质，即所有到某些相关结构的导子都是内导子或可以被内导子逼近。这些结构在被称为Banach代数的数学领域的研究中占有重要地位。对派生的研究起源于物理学，而我的研究(就像所有其他纯数学家的研究一样)是理论上的，主要着眼于知识的进步；它可能会在以后发现它在应用科学或工业中的应用。

We learn in a high school (or University) first calculus course that if f and g are two functions whose derivatives D(f) and D(g) exist, then the product function fg also has a derivative D(fg), which satisfies D(fg) = D(f)g + fD(g). In higher Mathematics there are structures comprised of elements-often called vectors-which can be added or multiplied, but, unlike the multiplication of functions, the multiplication of vectors of these structures, in general, may fail to satisfy fg=gf. Let's think of differentiation as being a mapping that assigns to every differentiable function another function (its derivative D(f)) subject to differentiation rules. There are similar mappings D on abstract structures that assign to each vector f in that structure a vector D(f) in the same or another related structure, subject to preservation of addition and multiplication by numbers, and act on the product of two vectors in the same fashion that differentiation does, i.e. D(fg) = D(f)g + fD(g). These mappings are called derivations. One of the simplest derivations is given by the formula D(f) = fg - gf. Here g is a fixed vector and f varies among the vectors of the given structure that D acts upon. These derivations are called inner derivations. We propose to find formulas for derivations on certain structures and to study those abstract structures that have the property that all the derivations into certain related structures are inner or can be approximated by inner derivations. These structures occupy a significant place in research in an area of Mathematics called Banach Algebras. The study of derivations originates from Physics and my research (like every other pure mathematician's research) is theoretical and primarily geared at advancement of knowledge; it may later find its use in applied sciences or industry.