Invariants for Completely Positive Maps and Multivariable Operator Theory

收件人：jjenkins@nsf.gov主题：摘要亲爱的乔，这是你要的摘要。如果这是你想要的，请告诉我。很抱歉这么晚才把这个还给你。因为你的纸条是在假期来的，我把它放在一边一段时间，然后就忘了！谢谢你的提醒。本项目涉及多变量算子理论。具体来说，我们关注的是几个变量多项式代数上的希尔伯特模块。我们为这些对象寻找具体的（数值的）不变量。其中最简单的是曲率不变量。希尔伯特模H的曲率不变量写成K(H)K(H)类似于偶数维黎曼流形的平均曲率，并且有一个渐近公式允许在许多情况下计算它的值。K(H)在我们所能确定的所有情况下都是一个整数，事实上，它符合希尔伯特模的高斯-博内定理的一种形式，这种形式承认“有限自由分辨率”，即K(H) = b(1) - b(2) + b(3) - b(4) +....式中b(1), b(2)，…为自由分辨率的贝蒂数。我们推测K(H)总是一个整数。不是每个希尔伯特模块（在我们拥有的类别中）都有有限的自由分辨率，我们正在开发一般计算K(H)的适当工具。量子理论中的时间流不同于经典物理学中的时间流，因为可观测的量不会相互交换。例如，海森堡关于一维量子系统的位置和动量观测的著名方程本质上是这样的：PQ - QP = 1。在过去的十年里，一个小而坚定的数学家小组一直在研究“E_0半群”理论。除此之外，这些数学对象给出了最简单的情况，描述了量子理论中时间流动的行为方式。当前的项目源于我们寻找区分不同类型E_0半群的方法的努力。这是通过计算与它们相关的特定数字来实现的，这些数字可以取不同的值。当一个人有两个数目不同的E_0半群时，就可以确定他有两个根本不同的系统。这个项目关注的是与E_0半群密切相关的简单对象相关的数值量的定义和计算。

To: jjenkins@nsf.gov Subject: abstract Dear Joe, Here is the abstract you requested. Please let me know if it's what you want. Sorry about the delay in getting this back to you. Because your note came during the holiday break, I put it aside for awhile, and then forgot about it! Thanks for sending the reminder. --Bill TECHNICAL DESCRIPTION This project concerns multivariable operator theory. Specifically, we are concerned with Hilbert modules over the algebra of polynomials in several variables. We seek concrete (numerical) invariants for such objects. The simplest of these is the curvature invariant. The curvature invariant of a Hilbert module H is written K(H). K(H) is analogous to the mean curvature of an even-dimensional Riemannian manifold, and there is an asymptotic formula which allows one to compute its value in many cases. K(H) has turned out to be an integer in all cases we have been able to decide, and in fact it obeys a form of the Gauss Bonnet theorem for the Hilbert modules which admit a "finite free resolution" in the following sense, K(H) = b(1) - b(2) + b(3) - b(4) +.... where b(1), b(2), ... are the Betti numbers of the free resolution. We conjecture that K(H) is always an integer. Not every Hilbert module (in the category we have) has a finite free resolution, and we are developing appropriate tools for computing K(H) in general. GENERAL DESCRIPTION The flow of time in quantum theory is different from the flow of time in classical physics, in that the observable quantities do not commute with each other. For example, Heisenberg's famous equation relating the position and momentum observables of a one-dimensional quantum system is essentially this: PQ - QP = 1. During the past ten years, a small but determined group of mathematicians has been working out the theory of "E_0 semigroups". Among other things, these mathematical objects give the simplest situations which describe the way the flow of time behaves in quantum theory. The current project has grown out of our efforts to find ways of distinguishing between different types of E_0 semigroups. This is accomplished by computing certain numbers associated with them that can take on different values. When one has two E_0 semigoups whose numbers are different, one can be assured that he has two systems which are fundamentally different. This project is concerned with the definition and calculation of numerical quantities which are associated to simpler objects that are closely related to E_0 semigroups.