Topological groups and flows, and associated function and operator algebras

我正在进行的研究集中在拓扑群，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。我对这些概念的研究使用了谐波分析、拓扑动力学和功能分析等强大工具。我研究的一个领域起源于本世纪初，哈拉尔德·玻尔关于实线上几乎周期函数的研究。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上的许多“几乎周期型”代数的研究。除了上面提到的工具，这项工作中的一个统一概念是G紧化的适当概念，就像Stone-Cech紧化一样，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方面起着重要作用。在其他正在进行的工作中，我研究了C*-代数（Hilbert空间上算子的闭代数），由算子方程(类似于UV =lambda VU，产生被广泛研究的“非理性旋转”C*代数的方程)以及这些代数与一些特殊群和流的联系；这项工作还有许多有趣的未完成之处。我工作的另一个重要方面是研究流和紧右拓扑群的例子、结构和其他性质。在这个领域一个显著的成功是在紧的右拓扑群上发现了Haar测度——也就是说，群上的一个概率测度是左右不变的，并且是唯一的；这一发现是与合著者J.S.皮姆共同完成的。

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.