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

我们在高中（或大学）第一堂微积分课上学到，如果f和g是两个函数，它们的导数D(f)和D(g)存在，那么乘积函数fg也有一个导数D(fg)，它满足D(fg) = D(f)g + fD(g)。在高等数学中，有一些由元素（通常称为向量）组成的结构，这些元素可以相加或相乘，但是，与函数的乘法不同，这些结构的向量的乘法通常不能满足fg=gf。我们把微分看成是一种映射它赋予每个可微函数另一个函数（它的导数D(f)）服从微分规则。在抽象结构上也有类似的映射D，将该结构中的每个向量f赋给相同或另一个相关结构中的向量D(f)，服从于保留数字的加法和乘法，并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。最简单的推导之一是公式D(f) = fg - gf。这里g是一个固定的向量，f随D作用的给定结构的向量而变化。这些推导称为内推导。我们建议寻找某些结构上的推导公式，并研究那些抽象结构，这些抽象结构具有某些相关结构的所有推导都是内的或可以被内推导近似的性质。这些结构在称为巴拿赫代数的数学领域的研究中占有重要地位。对推导的研究起源于物理学，我的研究（就像其他纯粹数学家的研究一样）是理论性的，主要是为了知识的进步；它以后可能会在应用科学或工业中得到应用。

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.