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      Weighted hypergroups and some questions in abstract harmonic analysis

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      ALAGHMANDAN-DISSERTATION.pdf (1.020Mb)
      Date
      2013-12-04
      Author
      Alaghmandan, Mahmood
      Type
      Thesis
      Degree Level
      Doctoral
      Metadata
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      Abstract
      Weighted group algebras have been studied extensively in Abstract Harmonic Analysis.Complete characterizations have been found for some important properties of weighted group algebras, namely, amenability and Arens regularity. Also studies on some other features of these algebras, say weak amenability and isomorphism to operator algebras, have attracted attention. Hypergroups are generalized versions of locally compact groups. When a discrete group has all its conjugacy classes finite, the set of all conjugacy classes forms a discrete commutative hypergroup. Also the set of equivalence classes of irreducible unitary representations of a compact group forms a discrete commutative hypergroup. Other examples of discrete commutative hypergroups come from families of orthogonal polynomials. The center of the group algebra of a discrete finite conjugacy (FC) group can be identified with a hypergroup algebra. For a specific class of discrete FC groups, the restricted direct products of finite groups (RDPF), we study some properties of the center of the group algebra including amenability, maximal ideal space, and existence of a bounded approximate identity of maximal ideals. One of the generalizations of weighted group algebras which may be considered is weighted hypergroup algebras. Defining weighted hypergroups, analogous to weighted groups, we study a variety of examples, features and applications of weighted hypergroup algebras. We investigate some properties of these algebras including: dual Banach algebra structure, Arens regularity, and isomorphism with operator algebras. We define and study Folner type conditions for hypergroups. We study the relation of the Folner type conditions with other amenability properties of hypergroups. We also demonstrate some results obtained from the Leptin condition for Fourier algebras of certain hypergroups. Highlighting these tools, we specially study the Leptin condition on duals of compact groups for some specific compact groups. An application is given to Segal algebras on compact groups.
      Degree
      Doctor of Philosophy (Ph.D.)
      Department
      Mathematics and Statistics
      Program
      Mathematics
      Supervisor
      Samei, Ebrahim; Choi, Yemon
      Committee
      Wang, Jiun-Chau; Szmigielski, Jacek; Hoffman, Sarah; Bickis, Mik
      Copyright Date
      November 2013
      URI
      http://hdl.handle.net/10388/ETD-2013-11-1298
      Subject
      Keyword 1: weighted hypergroup
      Keyword2 2: Leptin condition
      Keyword 3: Fourier algebra
      Keyword 4: Arens regular
      Keyword 5: Operator algebra
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