Operator algebras and hereditary subalgebras; visit of David Blecher.

这个项目涉及纯数学，特别是对所谓的算子代数的研究。“运算符”一词在这里指的是线性地图(如旋转、反射或放大)，而“代数”一词是指我们正在考虑一大批线性地图，而不是一次只考虑一个运算符--我们的线性地图集合在人们可以合理地使用线性地图所做的大多数事情下是封闭的，例如加减它们，将它们乘以常量，并组成其中的两个(进行一个又一个运算)。使算子代数成为算子代数的另一件事是，涉及的所有线性映射都作用在Hilbert空间上。粗略地说，希尔伯特空间是一个具有距离和角度概念的向量空间，毕达哥拉斯定律在其中成立。这就是我们似乎生活在的那种向量空间--毕达哥拉斯定律似乎确实适用于我们周围世界中的三角形。可以肯定的是，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，在球体的曲面上，人们可以很容易地画出一个三角形，其角度是三个直角；因此，毕达哥拉斯定律在这种情况下完全成立的可能性并不是很大。尽管如此，我们生活的世界似乎天生就是毕达哥拉斯式的，当人们研究另一种现代物理学--量子物理学时，这一点更是正确的。在那里，这个世界上的每个可观测量都是希尔伯特空间上的一个算子。量子物理学的微妙之处在其核心部分继承了深刻的几何性质，这些性质最终可以追溯到希尔伯特空间和毕达哥拉斯。例如，在下面隐藏的希尔伯特空间中，当系统的两个状态是垂直向量时，系统的两个状态是相互排斥的(作为生活中的替代)。如果每个可观测的量都是一个运算符，那么“可观测”的集合将是一个巨大的运算符集合。量子场论的一种表述要求，当你观察从特定空间区域观察到的量时，涉及的运算符形成了冯·诺伊曼代数。冯·诺伊曼代数可以粗略地认为是类固醇上的算子代数……因此，纯粹的数学家得出了所有这些结论，并希望将算子代数--线性映射作为一个整体--作为数学对象来研究。它们继承了与希尔伯特空间有关的一切事物的深层几何特征--例如，在单词“压缩近似恒等式”(“cai”是本研究的主要主题)中，“收缩”一词意味着我们看到的是从不增加两点之间距离的线性地图--旋转、反射和正交投影是收缩，但不是你放大后得到的线性地图--比方说，如果你把一切都放大了两倍。尽管如此，作为纯粹的数学家，我们仍然希望我们的线性地图是有界的，这意味着，虽然我们可以很高兴地将距离扩大两倍，但我们不允许单个操作符以任意大的数量扩大距离--但除了这一限制外，我们允许物理学家感兴趣的所有操作符。我们还研究了算子本身及其基本几何结构的深层复杂性。

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.