Banach algebras associated to locally compact groups

设 G 是一个群，即 G 是一个具有关联运算的集合，使得每个元素都有一个逆元。例如，G 可以被视为带有加法的正数或负数的集合，或者正交变换的集合，即线性变换后进行平移（仿射群）。我将考虑 G 配备一个拓扑，使得乘法和求逆继续进行。我将进一步假设拓扑是局部紧的，即由紧集组成的 G 的恒等式有邻域系统的基础。我未来几年的研究将包括：（a）与G相关的Banach代数（例如群代数、测度代数和傅里叶斯蒂尔切斯代数B（G））的几何、代数和拓扑性质的研究；b）由G的左正则表示生成的冯诺依曼代数VN（G）中相应非交换函数空间的对偶Banach代数的研究； (c) VN(G) 中的（非关联）Jordan 结构，用于 B(G) 中函数的不动点集或 B(G) 中的幂有界元素。我还将研究巴拿赫代数，它是冯·诺依曼代数的预解数，其中包括霍普夫·冯·诺依曼代数的预解数，特别是量子群代数。更具体地说，我将继续研究 G 的一些重要几何性质，例如： (A) G 上连续正定函数的闭子群的 Hahn-Banach 分离性质。 (B) 闭子群到全群上的连续正定函数的 Hahn-Banach 扩展性质。 (C) 不变补问题：局部紧群 G 的冯诺依曼代数 VN(G) 的每个弱*闭不变子空间是否都是不变补的？众所周知，局部紧群的情况就是这样，其恒等邻域的基础由紧集组成，这些紧集在内部自同构下或当 G 服从时是不变的。 我还将继续研究 (a) 利用紧化的深层组合和拓扑性质，自然数（或离散半群）的 Stone-Cech 紧化的测度代数； (b) 顺应性、遍历序列的弱收敛性和半群非扩张映射的不动点集的关系。更具体地说，我打算继续研究 Beurling 代数的第二对偶和正整数半群或更一般的取消半群的 Stone-Cech 紧致化的测度代数，利用半群紧致化的紧致右拓扑半群的深层结构、组合和拓扑性质。我还将研究 B(G) 的弱*闭子空间上的弱或弱*不动点属性，用于与连续表示和量子群代数相关的系数空间，这些系数空间依赖于它们的算子空间结构。

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.