Invariants for Completely Positive Maps and Multivariable Operator Theory

致：jjenkins@nsf.gov 主题：摘要 亲爱的 Joe， 这是您要求的摘要。 如果这是您想要的，请告诉我。 很抱歉延迟将此信息回复给您。 因为你的留言是在放假期间收到的，所以我把它搁置了一段时间，然后就忘记了！ 感谢您发送提醒。 --Bill 技术描述 该项目涉及多变量算子理论。 具体来说，我们关注多个变量多项式代数上的希尔伯特模。 我们为这些对象寻找具体的（数字）不变量。 其中最简单的是曲率不变量。 希尔伯特模 H 的曲率不变量写作 K(H)。 K(H) 类似于偶维黎曼流形的平均曲率，并且有一个渐近公式允许人们在许多情况下计算其值。 在我们能够确定的所有情况下，K(H) 都被证明是一个整数，事实上，它遵循希尔伯特模块的高斯邦尼定理的形式，该定理在以下意义上承认“有限自由分辨率”，K(H) = b(1) - b(2) b(3) - b(4) .... 其中 b(1)、b(2)、... 是自由分辨率的 Betti 数。 我们推测 K(H) 始终是整数。 并非每个希尔伯特模块（在我们拥有的类别中）都具有有限的自由分辨率，并且我们正在开发用于一般计算 K(H) 的适当工具。 一般描述 量子理论中的时间流动与经典物理学中的时间流动不同，因为可观测的量彼此不交换。 例如，海森堡关于一维量子系统的位置和动量可观测量的著名方程本质上是这样的：PQ - QP = 1。在过去的十年中，一小群但坚定的数学家一直在研究“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某些可以取不同值的数字来完成的。 当一个人拥有两个编号不同的 E_0 半组时，可以确信他拥有两个根本不同的系统。 该项目涉及与 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.