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 如果是的话请告诉我 你想要的 很抱歉这么晚才还给你。 因为你的信是在假期里寄来的，我把它放在一边一段时间，然后就忘了！ 感谢您发送 提醒 --比尔 技术说明 这个项目涉及多变量算子理论。 特别地，我们关注多项式代数上的Hilbert模 几个变量。 我们寻求这样的对象的具体（数值）不变量。 其中最简单的是曲率不变量。 希尔伯特模H的曲率不变量记为K（H）。 K（H）类似于偶数维的平均曲率 黎曼流形，有一个渐近公式， 在很多情况下都可以计算出它的价值。 K（H）已转 在所有的情况下，我们都能决定它是一个整数， 事实上它服从希尔伯特的高斯-邦纳定理的一种形式 模，其中承认一个“有限自由决议”在下面 感觉， K（H）= B（1）- B（2）+ B（3）- B（4）+. 其中B（1），B（2），.是自由分辨率的贝蒂数 我们猜想K（H）总是一个整数。 不是每个希尔伯特 模（在我们所拥有的范畴中）具有有限自由分解，并且 我们正在开发一般用于计算K（H）的适当工具。 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，因为可观察的量彼此不交换。 例如，海森堡著名的 位置和动量观测量之间的关系方程 一维量子系统基本上是这样的：PQ-QP = 1。 在过去的十年里，一个小而坚定的群体， 数学家们一直在研究“E_0半群”理论。 除此之外，这些数学对象给出了最简单的 描述时间流动方式的情况 量子理论 目前的项目是我们努力寻找区分不同类型E_0的方法的结果 半群 这是通过计算某些数字来实现的 与之相关的可以有不同的价值。 当一个 有两个E_0半群，它们的个数不同，其中一个可以是 他说，他有两个系统，这是根本不同的。 本项目涉及的定义和计算 与更简单的对象相关联的数值量， 与E_0半群密切相关。

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.