Mathematical Sciences: C*-Algebras and the Geometry of Hilbert Space

这是对两个著名的公开问题的数学研究，这些问题与算子代数上的纯态有关。状态这个术语来自于量子力学。在它的一些公式中，可观测用适当的算符代数中的自伴算符来表示，状态是算符代数上的某些线性泛函；因此，指定状态将为每个可观测分配一个数值。纯态是那些不是作为其他态的凸组合给出的态。从原则上讲，它们决定了代数的一切，但更准确地确定它们之间的联系导致了令人望而生畏的困难。本项目将解决的问题之一是将Stone-Weerstrass定理推广到一般的非交换情况，该定理可以被认为是关于交换算子代数的结果。另一个问题是证明Hilbert空间上所有有界算子的代数的极大对角子代数的纯态扩张的唯一性。

This is mathematical research in pursuit of two famous open problems having to do with pure states on operator algebras. The term state comes from quantum mechanics. In some formulations thereof, observables are represented by selfadjoint operators in an appropriate algebra of operators, and states are certain linear functionals on the operator algebra; thus, specifying a state assigns a numerical value to each of the observables. The pure states are those not given as convex combinations of other states. In principal, they determine everything about the algebra, but pinning down the connection more precisely has led to daunting difficulties. One of the problems the present project will address is that of extending the Stone-Weierstrass theorem, which may be thought of as a result about commutative operator algebras, to the general, noncommutative situation. The other problem is to show the uniqueness of pure state extensions from maximal diagonal subalgebras of the algebra of all bounded operators on a Hilbert space.