Invariants for Completely Positive Maps and Multivariable Operator Theory

致：jjenkins@nsf.gov主题：摘要亲爱的乔，这是您要的摘要。如果这是您想要的，请告诉我。抱歉，迟迟没能把这个还给你。因为你的纸条是在假期里寄来的，所以我暂时把它放在一边，然后就忘了！谢谢你发来的提醒。--法案技术描述本项目涉及多变量运算符理论。具体地说，我们研究了多元多项式代数上的Hilbert模。我们为这样的对象寻找具体的(数值)不变量。其中最简单的是曲率不变量。Hilbert模H的曲率不变量记为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可以在许多情况下计算它的值。K(H)已证明在我们所能确定的所有情况下都是一个整数，并且实际上它遵守Hilbert模的Gauss Bonnet定理的一种形式，在下面的意义下，K(H)=b(1)-b(2)b(3)-b(4)。其中b(1)、b(2)、..。是自由分解的Betti数。我们猜想K(H)总是一个整数。并不是每个Hilbert模(在我们已有的范畴中)都有有限自由分解，我们正在开发适当的工具来计算K(H)。一般描述量子理论中的时间流不同于经典物理学中的时间流，因为可观测量不相互交换。例如，海森伯格著名的关于一维量子系统的位置和动量观测的方程本质上是这样的：Pq-Qp=1。在过去的十年里，一小群坚定的数学家一直在研究“E_0半群”理论。在其他方面，这些数学对象给出了描述量子理论中时间流行为的最简单的情况。目前的项目是我们努力寻找区分不同类型的E_0半群的方法而发展起来的。这是通过计算与它们相关联的某些数字来实现的，这些数字可以采用不同的值。当一个人有两个数目不同的E_0半群时，可以肯定他有两个根本不同的系统。这个项目是关于与E_0半群密切相关的更简单对象的数值的定义和计算。

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.