Mathematical Sciences: The K-Theory and Isomorphism Invariants of Toeplitz Algebras

群作用于空间的情形，例如 作用于状态空间的一组随时间演化的真实的实数 在数学中经常会遇到 及其应用。 在物理学的例子中， 该系统的配置位于从远程设备流出的线路上， 从过去到遥远的未来 一个人对发生的事情感兴趣 总体而言，从长远来看，这可能是相当难以推断的， 仅仅是从短时间内发生的事情来判断。 为 分析的目的，可用的信息可以编码在 几种不同的方法：通过微分方程或向量 田野或叶理，或者像夏教授的 研究项目，由希尔伯特空间上的算子代数。 夏教授将研究一个相当一般的Toeplitz代数 最近开发的建筑（非 交换）动力系统;这与更多的 交叉乘积的概念。 的具体目标 他的工作是计算Toeplitz代数的K群， 他们的相关交换子理想，看看是否 Toeplitz代数是 底层动力系统

The situation of a group acting on a space, for instance the group of real numbers acting by time evolution on the state space of a physical system, is encountered quite often in mathematics and its applications. In the physics example, every possible configuration of the system lies on a line flowing from remote past to distant future. One is interested in what happens overall, in the long run, which may be quite difficult to infer solely from what happens over short intervals of time. For purposes of analysis, the available information can be encoded in several different ways: by differential equations or vector fields or foliations or, as is the case in Professor Xia's research project, by an algebra of operators on Hilbert space. Professor Xia will study a rather general Toeplitz algebra construction that has recently been developed for (non- commutative) dynamical systems; this is related to the more familiar notion of a crossed product. The specific objectives of his work are to calculate the K-groups of Toeplitz algebras and their associated commutator ideals, and to see whether the Toeplitz algebra is a complete isomorphism invariant of the underlying dynamical system.