Topological groups and flows, and associated function and operator algebras

我正在进行的研究主要围绕拓扑群、紧右拓扑群、流和C*-代数展开。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调和分析、拓扑动力学和泛函分析的强大工具。我的工作的一个领域起源于本世纪初，哈拉尔德·玻尔关于实线上的概周期函数的工作。从那时起，这门学科得到了极大的发展，现在包括了对群和半群G上的许多‘概周期型’代数的研究以及上面提到的工具，这项工作中的一个统一的概念是G的适当紧化概念，它类似于Stone-Cech紧化，只是它也考虑了G的代数结构。相关紧化的结构在决定代数和G的泛函分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，我研究了由算子方程(类似于UV=lambda vu，研究较多的无理旋转C*-代数的生成方程)，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的方面。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。这一领域的一个显著成功是在紧右拓扑群上发现了Haar测度--即群上的一种既是左不变的，也是唯一的概率度量；这一发现是与合著者J.S.PYM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.