Banach algebras associated to locally compact groups

与局部紧群相关的巴拿赫代数

• 批准号: RGPIN-2015-05520
• 负责人: Lau, AnthonyToMing
• 金额: $ 1.82万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=656724
• 关键词: Banach; algebras; associated; locally; compact

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.********

设G是一个群，即G是一个具有结合运算的集合，使得每个元素都有一个逆。 例如，G可以被认为是一组数字，正的或负的，加上，或一组正交变换，即一个线性变换后的平移（仿射群）。 我将考虑G配备一个拓扑，使得乘法和逆是连续的。进一步假设拓扑是局部紧的，即G的单位元的邻域系存在由紧集构成的基。我在今后几年的研究工作主要包括：（a）研究G（例如群代数、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）中相应的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）中幂有界元的VN（G）中的（非结合）Jordan结构.我还将研究Banach代数，它们是冯诺伊曼代数的预测，其中包括霍普夫冯诺伊曼代数的预测，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，如：*（A）G上连续正定函数闭子群的Hahn-Banach分离性 * (B)闭子群上连续正定函数到全群的Hahn-Banach扩张性质。(C)不变补问题：局部紧群G的群von Neumann代数VN（G）的每个弱 *-闭不变子空间是否不变补？ 这是已知的情况下，局部紧群的基础上的邻里的身份组成的紧集是不变的内部自同构，或当G是顺从的。* 我还将继续研究（a）自然数（或离散半群）的Stone-Cech紧化的测度代数，利用紧化的深层组合和拓扑性质;（B）非扩张映射半群的顺从性、遍历序列的弱收敛性与不动点集的关系。更具体地说，我打算继续研究Beurling代数的第二对偶和Stone-Cech紧化的正整数半群或更一般的可消半群的测度代数，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。我还将研究B（G）的弱 * 闭子空间上的弱或弱 * 不动点性质，其中系数空间与连续表示和量子群代数相关，量子群代数依赖于它们的算子空间结构。