Topological groups and flows, and associated function and operator algebras

我正在进行的研究是围绕拓扑群，紧右拓扑群，流和C*-代数。这些数学概念被数学家广泛研究;它们对物理学家、统计学家和社会科学家也非常有用。我的研究这些概念使用强大的工具，从谐波分析，拓扑动力学和功能分析。我的工作的一个领域起源于本世纪初，哈拉尔德·波尔（Harald Bohr）关于真实的线上的概周期函数的工作。从那时起，这门学科得到了巨大的发展，现在包括了对群和半群G上的“概周期型”代数的研究。除了上面提到的工具之外，本工作中的一个统一概念是G的紧化的适当概念，它类似于Stone-Cech紧化，除了还考虑了G的代数结构。相关紧化的结构在确定代数和G的泛函、解析和动力学性质中起着重要的作用。 在其他正在进行的工作中，我研究C*-代数（封闭代数的运营商希尔伯特空间）产生的运营商方程（类似于UV =lambda VU，方程产生的研究'无理旋转' C*-代数）和连接这些代数与一些特殊的群体和流动;有许多有趣的未完成的方面这项工作。 我工作的另一个重要方面是研究流和紧致右拓扑群的例子、结构和其他性质。一个显着的成功，在这一领域是发现哈尔措施紧右拓扑群-也就是说，一个概率措施的一组，这是左和右不变的，唯一的，因为这样;这一发现是在联合工作与合著者J. S。皮姆

My ongoing research is centred around topological groups, compact right topological groups, flows and C*-algebras. These mathematical concepts are widely studied by mathematicians; they are also very useful to physicists, statisticians and social scientists. My study of these concepts uses powerful tools from harmonic analysis, topological dynamics and functional analysis. One area of my work has its origins at the beginning of this century, in the work of Harald Bohr on almost periodic functions on the real line. Since then the subject has grown enormously and now includes the study of many algebras of 'almost periodic type' on groups and semigroups G. As well as the tools mentioned above, a unifying concept in this work is the appropriate notion of compactification of G, which is like the Stone-Cech compactification, except that account is taken also of the algebraic structure of G. The structure of the relevant compactifications plays an important role in determining functional analytic and dynamical properties of the algebras and of G.    In other ongoing work, I study C*-algebras (closed algebras of operators on Hilbert space) generated from operator equations (analogous to UV =lambda VU, the equation generating the much -studied 'irrational rotation' C*-algebras) and the connection of these algebras with some special groups and flows; there are many interesting unfinished aspects of this work.    A further important aspect of my work is the study of examples, structure and other properties of flows and compact right topological groups. A notable success in this area was the discovery of Haar measure on compact right topological groups - that is, a probability measure on the group that is both left and right invariant, and unique as such; this discovery was made in joint work with coauthor J.S. Pym.