26 research outputs found
Amenability notions of hypergroups and some applications to locally compact groups
Different notions of amenability on hypergroups and their relations are
studied. Developing Leptin's theorem for discrete hypergroups, we characterize
the existence of a bounded approximate identity for hypergroup Fourier
algebras. We study the Leptin condition for discrete hypergroups derived from
the representation theory of some classes of compact groups. Studying
amenability of the hypergroup algebras for discrete commutative hypergroups, we
obtain some results on amenability properties of some central Banach algebras
on compact and discrete groups.Comment: Significant revisions to the paper. Abstract revised, some typos
corrected, some references added. The exposition has been improved while the
paper has been shortened significantly. An error in the proof of the Leptin
theorem is corrected by restricting the case to discrete hypergroup
Approximate amenability of Segal algebras II
We prove that every proper Segal algebra of a SIN group is not approximately
amenable
Weighted discrete hypergroups
Weighted group algebras have been studied extensively in Abstract Harmonic
Analysis where complete characterizations have been found for some important
properties of weighted group algebras, namely amenability and Arens regularity.
One of the generalizations of weighted group algebras is weighted hypergroup
algebras. Defining weighted hypergroups, analogous to weighted groups, we study
Arens regularity and isomorphism to operator algebras for them. We also examine
our results on three classes of discrete weighted hypergroups constructed by
conjugacy classes of FC groups, the dual space of compact groups, and
hypergroup structure defined by orthogonal polynomials. We observe some
unexpected examples regarding Arens regularity and operator isomorphisms of
weighted hypergroup algebras.Comment: 27 pages. This version is shorter but still covers all the main
results of the previous on
Amenability properties of the central Fourier algebra of a compact group
We let the central Fourier algebra, ZA(G), be the subalgebra of functions u
in the Fourier algebra A(G) of a compact group, for which u(xyx^{-1})=u(y) for
all x,y in G. We show that this algebra admits bounded point derivations
whenever G contains a non-abelian closed connected subgroup. Conversely when G
is virtually abelian, then ZA(G) is amenable. Furthermore, for virtually
abelian G, we establish which closed ideals admit bounded approximate
identities. We also show that if ZA(G) is weakly amenable, even
hyper-Tauberian, exactly when G admits no non-abelian connected subgroup. We
also study the amenability constant of ZA(G) for finite G and exhibit totally
disconnected groups G for which ZA(G) is non-amenable.Comment: 21 pages, mildy expository improvements mad
Fourier algebras of hypergroups and central algebras on compact (quantum) groups
This paper concerns the study of regular Fourier hypergroups through
multipliers of their associated Fourier algebras. We establish hypergroup
analogues of well-known characterizations of group amenability, introduce a
notion of weak amenability for hypergroups, and show that every discrete
commutative hypergroup is weakly amenable with constant 1. Using similar
techniques, we provide a sufficient condition for amenability of hypergroup
Fourier algebras, which, as an immediate application, answers one direction of
a conjecture of Azimifard--Samei--Spronk [J. Funct. Anal. 256(5) 1544-1564,
2009] on the amenability of for compact groups . In the final
section we consider Fourier algebras of hypergroups arising from compact
quantum groups , and in particular, establish a completely
isometric isomorphism with the center of the quantum group algebra for compact
of Kac type.Comment: 23 page
Weighted hypergroups and some questions in abstract harmonic analysis
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