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Closed ideals of the Banach algebra of bounded operators on a Banach space

Banach 空间上有界算子的 Banach 代数的封闭理想

基本信息

批准号：
EP/F023537/1
负责人：
Niels Laustsen
金额：
$ 19.7万
依托单位：
Lancaster University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2008
资助国家：
英国
起止时间：
2008 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FF023537%2F1
关键词：
Closed ideals Banach algebra bounded

项目摘要

We carry out basic research into a mathematical object called a Banach algebra. To explain what this is, think of the set of integers {...,-2,-1,0,1,2,...}. You can add, subtract and multiply two integers, and you can measure the distance between them; for instance, 2+3=5, 2-3=-1, and 2x3=6, and the distance between 2 and 3 is 1. A Banach algebra shares all these properties: its elements can be added, subtracted, and multiplied together, and you can measure the distance between them.A fundamental property of the integers is that every integer (apart from 0 and +/-1) can be written as a product of prime numbers. (Recall that a number is prime if it can only be divided by 1 and itself.) Consequently, we think of the prime numbers as the building blocks of the integers; many questions about integers can be answered by considering prime numbers first and then generalizing to all integers by extending to products of primes. In a Banach algebra, the objects playing the role of building blocks are called ideals. (In this context the word ideal has no relation to its standard usage.) The purpose of this project is to determine all the ideals of certain Banach algebras, that is, describe their building blocks; this knowledge will be useful in future research, just as prime numbers are useful when studying the integers.The project focuses on ideals for a particular kind of Banach algebra, namely Banach algebras of operators. You can think of an operator as a mathematical machine which takes some input at one end, processes it, and then delivers an output at the other end. You multiply two such operators by using the output of the first operator as input for the second. This multiplication has a very important special feature: the order in which you multiply two operators matters, that is, when A and B are operators, A times B may give a different result from B times A. We refer to this fact by saying that operators do not commute. Of course, this phenomenon has no counterpart among the integers; 2 times 3 and 3 times 2 are always equal! Thus at first sight it may seem rather strange that operators do not commute, but it is not - we see similar things happen every day; for instance, when you put on socks and shoes, the order is essential.The fact that operators do not commute has a profound influence on the Banach algebras which we study and gives the subject a very different flavour from that of the integers. Importantly, this difference is also the driving force behind the most significant application of operators in the physical world. When quantum mechanics was founded in the 1920's, the physicist Heisenberg stated as a basic principle that, at atomic level, you cannot simultaneously know both the precise speed and position of a particle. This is of course in stark contrast to our everyday experience, where we usually know both where we are and how fast we are going when driving a car, say. Heisenberg's claim led him to propose that physical quantities like position and speed should not be represented by numbers (or functions), but by operators; the fact that certain operators do not commute explains why the corresponding quantities cannot be known simultaneously. Heisenberg's use of operators in quantum mechanics was elaborated on by von Neumann in the 1930's, giving the subject a solid mathematical grounding. This work laid the foundations of a new research area, operator algebras, which has flourished ever since.In 1940 Calkin (a student of von Neumann's) gave the first complete description of the ideals of a Banach algebra of operators. Similar results for two other Banach algebras of operators were obtained in the 1960's, but it then took until 2004 before the next such complete description appeared, in joint work of Loy, Read and myself. This discovery has sparked new activity in the area, with recent results by Daws and by Schlumprecht, Zsak and myself in a collaboration which we plan to continue through this project.

我们对一个叫做巴拿赫代数的数学对象进行基础研究。为了解释这是什么，想想整数的集合{...，- 2，-1，0，1，2，.}。你可以对两个整数进行加、减、乘运算，也可以测量它们之间的距离;例如，2+3=5，2-3=-1，2x 3 =6，2和3之间的距离是1。一个Banach代数共享所有这些属性：它的元素可以加，减，乘在一起，你可以测量它们之间的距离。整数的一个基本属性是，每个整数（除了0和+/-1）可以写为素数的乘积。（回想一下，如果一个数只能被1和它本身整除，那么它就是素数。）因此，我们认为素数是整数的基石;许多关于整数的问题可以通过首先考虑素数，然后通过扩展到素数的乘积来推广到所有整数来回答。在巴拿赫代数中，扮演积木角色的对象称为理想。(In在本文中，理想这个词与它的标准用法没有关系。）这个项目的目的是确定某些Banach代数的所有理想，也就是说，描述它们的构建块;这些知识将在未来的研究中很有用，就像素数在研究整数时很有用一样。这个项目的重点是一种特殊类型的Banach代数的理想，即Banach代数的算子。你可以把运算符看作是一台数学机器，它在一端接受一些输入，处理它，然后在另一端提供一个输出。通过使用第一个运算符的输出作为第二个运算符的输入，可以将两个这样的运算符相乘。这种乘法有一个非常重要的特殊功能：两个运算符相乘的顺序很重要，也就是说，当A和B是运算符时，A乘以B可能得到与B乘以A不同的结果。我们指的是这一事实，说运营商不通勤。当然，这种现象在整数中没有对应的; 2乘以3和3乘以2总是相等的！因此，乍一看似乎很奇怪，运营商不交换，但它不是-我们看到类似的事情发生的每一天，例如，当你穿上袜子和鞋子，秩序是必不可少的。事实上，运营商不交换有深远的影响，我们研究的Banach代数，并给予该主题一个非常不同的味道，从整数。重要的是，这种差异也是操作符在物理世界中最重要的应用背后的驱动力。当量子力学在20世纪20年代成立时，物理学家海森堡指出，在原子水平上，你不能同时知道粒子的精确速度和位置。这当然与我们的日常经验形成鲜明对比，我们通常知道我们在哪里，以及我们开车时的速度。海森堡的主张使他提出，像位置和速度这样的物理量不应该用数字（或函数）来表示，而是用算子来表示;某些算子不对易的事实解释了为什么相应的量不能同时被知道。海森堡在量子力学中使用的算符在20世纪30年代由冯·诺依曼详细阐述，给了这个主题坚实的数学基础。这项工作奠定了基础的一个新的研究领域，算子代数，这一直蓬勃发展至今。在1940年卡尔金（学生冯诺依曼）了第一次完整的描述理想的Banach代数的运营商。类似的结果为其他两个巴拿赫代数的运营商获得在20世纪60年代，但直到2004年才出现下一个这样完整的描述，在联合工作的洛伊，阅读和我自己。这一发现引发了该领域的新活动，最近的成果由道斯和Schlumprecht，Zsak和我自己合作，我们计划通过这个项目继续下去。