Algebras of abstract harmonic analysis

群的数学概念是对称性的最基本表达。像正方形这样的物体只允许有限多个对称，而球体或平面允许无限多个对称。为了使这样一组对称的分析更容易处理，并记住更多底层球体或平面的空间特征，比方说，需要处理对称群的空间属性，这是我们称为拓扑的概念。从而得到拓扑群。此外，大多数自然的例子，如上面的例子，都表明在单个点附近有一定的有限性，我们称之为局部紧性。 对于这些局部紧群，重要的是要了解它们的表示，即它们如何作用于具有拓扑的其他集合。Banach代数的使用是封装和帮助理解这些操作的工具。这类代数，包括群代数和测度代数，弱概周期函数，傅立叶代数和傅立叶-斯泰尔特斯代数，以及相关的西格尔和比尔林代数，都在这一分析中扮演着有趣而独特的角色。我的研究主要集中在局部紧群的结构和伴随的Banach代数的结构之间的相互作用。我最基本的问题是：给定一类局部紧群的性质，这个性质在相关的Banach代数中是如何证明的，反之亦然。这些问题的解决往往迫使我们在这些Banach代数上使用不明显的结构，如算子空间结构。例如，顺从性是群上的一个基本的平均性质，只有在具有算子空间平均性质的傅里叶代数中才能看到，而不是通过只能用Banach代数技巧才能看到的性质来证明。 物理学家可以观察到的一些对称性需要更微妙的对称性。值得注意的是，这些被称为量子群的数学公式也概括了广义的庞特里亚金对偶，这是理解群代数和傅立叶代数相互关系的框架。因此，这个更一般的框架也引起了我的一些研究注意。

The mathematical concept of a group is the most basic expression of symmetry. An object like a square admits only finitely many symmetries, whereas a sphere or plane admits infinitely many. In order to keep the analysis of such a group of symmetries more tractable, and to remember more of the spatial features of the underlying sphere or plane, say, one needs to deal with the spatial properties of the symmetry group, a concept we call topology. Hence we obtain topological groups. Furthermore, most natural examples, such as those above, suggest a certain finiteness near individual points, a condition we call local compactness. For these locally compact groups it is important to understand their representations, i.e. how they act on other sets with topology. The use of Banach algebras is a tool to encapsulate and help to understand these actions. Such algebras, which include group and measure algebras, weakly almost periodic functions, Fourier and Fourier-Steiltjes algebras, and associated Segal and Beurling algebras all play interesting and distinctive roles in this analysis. My research focusses on the interplay of the structures of locally compact groups and structures of the associated Banach algebras. My most basic question: given a property of a class of locally compact groups, how is that property witnessed in the associated Banach algebras, and vice versa. The resolutions of these questions often force us to use inobvious structures on these Banach algebras, such as operator space structures. For example, amenability,  a fundamental averaging property on groups, is witnessed only in Fourier algebras with an operator space averaging property, and not by a property which can be seen only with Banach algebra techniques. Some symmetries observable to physicists require much more subtle symmetries. Remarkably, the mathematical formulation of these, known as quantum groups, also encapsulates generalized Pontryagin duality, which is a framework for understanding the group algebra and Fourier algebra in relation to one another. Hence this more general framework also catches some of my research attention.