Operator algebras and hereditary subalgebras; visit of David Blecher.

算子代数和遗传子代数；

• 批准号: EP/K019546/1
• 负责人: C J Read
• 金额: $ 2.34万
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FK019546%2F1
• 关键词: Operator; algebras; hereditary; subalgebras; visit

This project concerns Pure Mathematics, in particular the study of what are called Operator Algebras. The word "operator" here means a linear map (such as a rotation, a reflection or an enlargement), and the word "algebra" means that we are considering a large collection of linear maps en masse, rather than just one operator at a time - our collection of linear maps to be closed under most of the things one can sensibly do with linear maps, such as adding and subtracting them, multiplying them by constants, and composing two of them (doing one operation after another). The other thing that makes an operator algebra an operator algebra is the fact that all the linear maps involved are acting on a Hilbert space. A Hilbert space is, loosely speaking, a vector space with a notion of distance and a notion of angle in which Pythagoras' law holds true. That is the kind of vector space in which we seem to be living - Pythagoras' law really does seem to hold for triangles in the world around us. To be sure, we reckon that this is not quite precisely correct - since the advent of the general relativity, we believe that space is curved - and it is not hard to see that on the curved surface of a sphere, one can easily draw a triangle whose angles are three right angles; so the chances of Pythagoras' law holding perfectly in that case are not so good. Nonetheless the world we live in seems to be inherently Pythagorean, and this is even more true when one investigates the other modern physics, the quantum physics. There, every observable quantity in this world is an operator on a Hilbert space. The subtleties of the quantum physics inherit deep geometric properties at their heart which in the end, go back to Hilbert spaces and Pythagoras. For example, two states of the system are mutually exclusive (as alternatives in life) when they are orthogonal vectors, at right angles in the hidden Hilbert space underneath. And if every observable quantity is to be an operator, the collection of "observables" is going to be a large collection of operators. One formulation of the quantum field theory requires that when you look at the quantities you can observe from a particular region of space, the operators involved form a von Neumann algebra. A von Neumann algebra might loosely be thought of as an operator algebra on steroids... So the pure mathematician comes to all this and wants to study the operator algebras - the linear maps en masse - as mathematical objects in their own right. They inherit the deep geometrical flavour of everything that has to do with Hilbert spaces - for example, in the words "contractive approximate identity", ("cais" are a main topic of this present research), the word "contractive" means we look at linear maps which never increase the distance between two points - rotations, reflections and orthogonal projections are contractions, but not the linear maps that you get if you enlarge - if, say, you scale everything up by a factor of two. As pure mathematicians we nonetheless like our linear maps to be bounded, which means that while we might happily enlarge by a factor of two, we do not allow a single operator to enlarge distances by arbitrarily large amounts - but apart from that one restriction we allow all the operators that the physicist is interested in. And we study the deep complexity of the operators themselves and their underlying geometry.

这个项目涉及纯数学，特别是研究什么是所谓的算子代数。这里的“算子”一词是指线性映射（如旋转，反射或放大），而“代数”一词意味着我们正在考虑大量的线性映射，而不是一次只考虑一个运算符-我们的线性映射集合在大多数人们可以合理地对线性映射做的事情下是封闭的，如加和减它们，乘以常数，两个人，一个人，一个人，一个人，一个人。 使算子代数成为算子代数的另一件事是，所有涉及的线性映射都作用在希尔伯特空间上。广义地说，希尔伯特空间是一个具有距离和角度概念的向量空间，其中毕达哥拉斯定律成立。这就是我们似乎生活在其中的向量空间--毕达哥拉斯定律似乎确实适用于我们周围世界的三角形。当然，我们认为这并不完全正确--自从广义相对论出现以来，我们相信空间是弯曲的--不难看出，在球面的曲面上，人们可以很容易地画出一个三角形，其角是三个直角;因此，在这种情况下，毕达哥拉斯定律完全成立的可能性并不太大。尽管如此，我们生活的世界似乎本质上是毕达哥拉斯的，当我们研究另一个现代物理学--量子物理学时，情况就更是如此了。在那里，这个世界上的每一个可观测量都是希尔伯特空间上的算子。量子物理学的微妙之处在其核心继承了深刻的几何性质，最终可以追溯到希尔伯特空间和毕达哥拉斯。例如，当系统的两个状态是正交向量时，它们是相互排斥的（作为生活中的替代品），在隐藏的希尔伯特空间中呈直角。如果每个可观测量都是一个算子，那么“可观测量”的集合将是一个很大的算子集合。量子场论的一个公式要求，当你从一个特定的空间区域观察你能观察到的量时，所涉及的算子形成了一个冯·诺依曼代数。一个冯诺依曼代数可能会松散地被认为是一个运营商代数类固醇.因此，纯数学家来到这一切，并希望研究算子代数-线性映射-作为数学对象本身。它们继承了与希尔伯特空间有关的一切事物的深刻几何味道-例如，在“压缩近似恒等式”中，（“CAIS”是本研究的一个主要主题），“收缩”一词意味着我们观察从不增加两点之间距离的线性映射-旋转、反射和正交投影都是收缩，但不是你放大后得到的线性地图--如果，比如说，你把所有东西都放大一倍。作为纯数学家，我们仍然希望我们的线性映射是有界的，这意味着，虽然我们可能很高兴地将距离放大两倍，但我们不允许单个算子将距离放大任意大的量--但除了这一限制，我们允许物理学家感兴趣的所有算子。我们还研究了算子本身及其基本几何的深层复杂性。