Banach algebras associated to locally compact groups

设G是一个群，即G是一个集合，它的关联运算使得每个元素都有一个逆。例如，G可以被认为是一组正的或负的数字，加上加法，或者是一组正交变换，即一个线性变换后的平移（仿射群）。我将考虑G配备了这样的拓扑，即乘法和反转是连续的。我进一步假设该拓扑是局部紧致的，即存在由紧致集组成的G的恒等式的邻域系统的基。我在未来几年的研究将包括：(a)研究与G相关的Banach代数的几何、代数和拓扑性质（如群代数、测度代数和Fourier Stieltjes代数B(G)）；b)研究由G的左正则表示生成的von Neumann代数VN(G)中相应非交换函数空间的对偶Banach代数；(c) B(G)中函数的不动点集或B(G)中的幂有界元的不动点集在VN(G)中的（非结合的）Jordan结构。我也会学习Banach代数，它是冯·诺伊曼代数的前偶，包括Hopf冯·诺伊曼代数的前偶，特别是量子群代数。更具体地说，我将继续研究G的一些重要的几何性质，例如：

Let G be a group, i.e. G is a set with an associative operation such that each element has an inverse. For example, G may be taken to be the set of numbers, positive or negative, with addition, or the set of orthogonal transformations i.e. a linear transformation followed by translation (the affine group). I will consider G equipped with a topology such that multiplication and inversion are continues. I will further assume that the topology is locally compact, that is there is basis for the neighborhood system for the identity of G consisting of compact set. My research in the next few years will consist of : (a) the study of geometric, algebraic and topological properties on Banach algebras associated for a G (e.g. group algebra, measure algebra and the Fourier Stieltjes algebra B(G); b) the study of dual Banach algebras of the corresponding non-commutative function space in the von Neumann algebra VN(G) generated by the left regular representation of G; (c) the (non-associative) Jordan structure in VN(G) for the fixed point set of a function in B(G) or power bounded elements in B(G). I will also study Banach algebras which are preduals of von Neumann algebras which will include preduals of Hopf von Neumann algebras, in particular quantum group algebras. More specifically, I will continue to study some important geometric properties of G such as: