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，如：***(A) G上连续正定函数对闭子群的Hahn-Banach分离性质***(B)闭子群上连续正定函数对满群的Hahn-Banach可拓性质。****(C)不变补问题：局部紧群G的群von Neumann代数VN(G)的每一个弱*闭不变子空间是否不变补？这是已知的，对于具有由紧集组成的紧群的邻域基的局部紧群的情况，这些紧群在内自同构下是不变的，或者当G是可服从的。******我还将继续研究(a)使用紧化的深度组合和拓扑性质的自然数（或离散半群）的stone - ech紧化的度量代数；(b)非扩张映射半群遍历序列的适应性、弱收敛性与不动点集的关系。更具体地说，我打算利用半群紧化的紧右拓扑半群的深层结构、组合和拓扑性质，继续研究Beurling代数的第二对偶和正整数半群或更一般地说是可消半群的stone - ech紧化的测度代数。我还将研究B(G)的弱*闭子空间上的弱或弱*不动点性质，用于与连续表示和依赖于它们的算子空间结构的量子群代数相关的系数空间。********

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:***(A) The Hahn-Banach separation property for closed subgroups by continuous positive definite functions on G.***(B) The Hahn-Banach extension property for a continuous positive definite functions on a closed subgroup to the full group.****(C) The invariant complementation problem: Is every weak*-closed invariant subspace of the group von Neumann algebra VN(G) of a locally compact group G invariantly complemented? This is known to be the case for locally compact groups with a basis of neighborhoods of the identity consisting of compact sets which are invariant under inner automorphisms, or when G is amenable. ******I will also continue to study (a) the measure algebra of the Stone-Cech compactification of the natural number (or discrete semi-group) using deep combinatorial and topological properties of the compactification; (b) the relation of amenability, weak convergence of ergodic sequences and fixed point set for semi-group of non-expansive mappings. More specifically, I intend to continue to study the second dual of Beurling algebras and the measure algebra of the Stone-Cech compactification of the semigroup of positive integer or more generally a cancellative semigroup using the deep structure, combinatorial and topological properties of the compact right topological semigroup of the compactification of the semigroup. I will also study weak or weak* fixed point properties on weak* closed subspaces of B(G) for coefficient spaces associated to a continuous representation and quantum group algebras which are dependent on their operator space structures.********