Banach algebras associated to locally compact groups

点击翻译按钮获取中文摘要

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.********