Where to find the image of a derivation
Where to find the image of a derivation
复制标题
哪里可以找到推导图像
DOI：
10.4064/-30-1-237-249
复制
发表时间：
1994
期刊：
影响因子：
--
通讯作者：
Martin Mathieu
中科院分区：
文献类型：
--
作者：
Martin Mathieu
在这篇论文中，我们打算提供一些关于Banach代数的无界导数仍然无法解的问题，非交换Singer-Wermer猜想的最新工作的概述。特别地，我们讨论了包含图像中拟不完备性的导数的几个全局和局部性质。1. 去哪里找……如果用巴拿赫代数来建模，推导可以作为物理系统可逆演化的生成器。不仅在历史上，这一观点有力地推动了对衍生的研究，以及它们的性质与巴拿赫代数结构的关系。最简单的例子是内导δa: x7→xa−ax，其中a是巴拿赫代数a中的一个给定元素，人们可能会认为没有其他元素。事实上，一个人在道德上是正确的：一个不消失的第一上同调群通常被认为是一种障碍，而不是一种乐趣，在哲学上也是如此：在一个更大的巴拿赫代数中，推导往往是内在的。这里有一个例子。假设A是单位的。（因为每一个导数δ在恒等上都消失了，所以这不是对普遍性的限制，因此将被默认。）将A与A上所有有界线性算子的代数B的闭子代数通过左正则表示a7→La识别，则有[La， δ](x) = aδ(x)−δ(ax) =−δ(A) x = L−δ(A) (x) (x∈A)，即[La， δ] = L−δ(A)，或者在上述识别下，对于所有A∈A， δ(A) = δ−δ(A)。当然，为此我们必须假设δ是有界的。这已经表明，实际的问题是无界的推导，但众所周知，即使有界的推导一般也不需要在严格意义上是内的。二级46-02、46H99、47-02。这篇论文已成定稿，不会在其他地方发表任何版本。
With this paper, we intend to provide an overview of some recent work on a problem on unbounded derivations of Banach algebras that still defies solution, the noncommutative Singer–Wermer conjecture. In particular, we discuss several global as well as local properties of derivations entailing quasinilpotency in the image. 1. Where to look for . . . Derivations may serve as the generators of reversible evolutions of a physical system, say, if this is modelled by a Banach algebra. Not only historically, this point of view gave a strong impetus to the investigation of derivations and of how their properties relate to the structure of Banach algebras. The easiest examples one encounters are the inner derivations δa : x 7→ xa − ax, where a is a given element in the Banach algebra A, and one may be tempted to think that there are no others. Indeed, one is morally right: a non-vanishing first cohomology group is generally considered rather as an obstacle than a delight, as well as philosophically: often derivations become inner in a larger Banach algebra. Here is an example. Let us suppose that A is unital. (Since every derivation δ vanishes on the identity, this is no restriction of generality and will therefore be tacitly assumed henceforth.) Identifying A with a closed subalgebra of the algebra B of all bounded linear operators on A via the left regular representation a 7→ La, one has [La, δ](x) = aδ(x)− δ(ax) = −δ(a)x = L−δ(a)(x) (x ∈ A) , that is, [La, δ] = L−δ(a) or, under the above identification, δ(a) = δ−δ(a) for all a ∈ A. Of course, to this end we have to assume that δ is bounded. This already indicates that the actual problem is with unbounded derivations, but as is well known, even bounded derivations in general need not be inner in the strict sense 1991 Mathematics Subject Classification: Primary 47B47; Secondary 46-02, 46H99, 47-02. The paper is in final form and no version of it will be published elsewhere.