Banach algebra and operator space techniques in topological group theory

我们的研究背后有两条主要线索。一种是经典的“调和分析”，在这种分析中，函数被“分成周期部分”。例如，声波可以被分成不同的频率，一旦人们在“频域”中工作，就有可能处理信号--例如，可以去除人耳无法处理的频率，这是支持现代数字音频压缩的思想之一。这一思想可以追溯到近300年前的傅里叶，他说：“每个函数都可以写成三角函数的无穷和”。这一思想的发展是随后几个世纪许多分析的基础。在适当的时候，在20世纪，调和分析成为一个庞大而复杂的学科；它通过检查这些对象的‘傅立叶变换’来研究一般的‘局部紧群’上的函数和度量。我们的主题后面的第二股是希尔伯特空间上的有界线性算子；这些算子将作用于有限多维欧几里德空间的矩阵的思想推广到无限多维。希尔伯特空间在调和分析中自然产生：例如，一个声波有无限多个频率，每个频率在“频率空间”中产生一个维度。人们很自然地认为，在频率空间上的合理操作将是线性的--例如，添加声波的概念。希尔伯特空间上的算符理论在量子力学的数学基础上也有许多应用，历史上就是在这一领域产生的。如今，这个数学领域被称为‘算符代数’的研究，是一个广泛的研究领域，跨越纯数学和应用数学。在过去的15-20年里，算符代数和调和分析之间的新联系出现在‘算符空间’理论中。它使用Hilbert空间上的有界线性算子空间作为空间和代数的模型。这种物体之间的自然映射是“完全有界的算子”。这样的地图似乎挑出了所有有界算符集合中的一个重要子类。现在，这门学科被应用于现代调和分析；已经表明，许多单独开发的结果实际上是最近发现的更一般原理的特例。我们研讨会的目的是将本学科这两个方面的约30名杰出专家聚集在一起，进行一段时间的密集学习和演讲，面向其他专家和其他数学领域的研究生和同事。许多英国人将参加这些讲座。我们期望现有的数学家小组之间的合作得到巩固，并将形成新的合作；这些合作将在研讨会本身和未来产生一些我们已知的公开问题的解决方案，并允许参与者使用他们在研讨会上学到的想法来形成新的综合和方法。这些新成果将在开放获取的预印本服务器和国际研究期刊上传播。

There are two major strands lying behind our research. One is that of classical `harmonic analysis', in which functions are `separated into their periodic parts'. For example, a sound wave can be split into different frequencies, and once one is working in the `frequency domain' it is possible to process the signal-- for example, frequencies which the human ear cannot process can be removed, one of the ideas underpinning modern digital audio compression. This idea goes back nearly 300 hundred years to Fourier, who stated that `each function can be written as the infinite sum of trigonometric functions'. The developments of this idea lie at the base of much analysis in the subsequent centuries. In due course, in the 20th century, the subject of harmonic analysis became a huge and sophisticated subject; it considers functions and measures on general `locally compact groups' by inspecting the `Fourier transform' of these objects.The second strand behind our subject is that of bounded linear operators on Hilbert spaces; these operators generalize to infinitely many dimensions the idea of a matrix acting on a Euclidean space of finitely many dimensions. Hilbert spaces arise naturally in harmonic analysis: for example, a sound wave has infinitely many frequencies, each frequency giving rise to a dimension in `frequency space'. It is natural to expect that a sensible operation on frequency space will be linear-- the notion of adding sound waves, for example. The theory of operators on a Hilbert space also has numerous applications in the mathematical foundations of Quantum Mechanics, and historically arose in this area. These days this area of mathematics is refered to as the study of `operator algebras', and is a vast area of research with links across pure and applied mathematics.In the last 15--20 years a new link between operator algebras and harmonic analysis has arisen in the theory of `operator spaces'. This uses spaces of bounded linear operators on Hilbert spaces as a model for spaces as well as algebras. The natural maps between such objects are the `completely bounded operators'. Such maps seem to single out an important subclass of the collection of all bounded operators.Now this subject is being applied to modern harmonic analysis; it has already been shown that many results developed separately are really special cases of more general principles that have been discovered recently.It is the purpose of our workshop to bring together about 30 distinguished experts inboth aspects of our subject for a period of intensive study and lectures, intended for both the other experts and for graduate students and colleagues in other areas of mathematics. A number of UK-based people will attend these lectures.We expect existing collaborations between small groups of mathematicians to be consolidated and that new ones will form; these collaborations will generate, both within the workshop itself and in the future, some solutions to the open problems that we know of, and allow the participants to form new syntheses and approaches using ideas that they learn at the workshop. These new results will be disseminated in open access preprint servers and in international research journals.