Dual Space and Hyperdimension of Compact Hypergroups

Abstract

We characterize dual spaces and compute hyperdimensions of irreducible representations for two classes of compact hypergroups namely conjugacy classes of compact groups and compact hypergroups constructed by joining compact and finite hypergroups. Also studying the representation theory of finite hypergroups, we highlight some interesting differences and similarities between the representation theories of finite hypergroups and finite groups. Finally, we compute the Heisenberg inequality for compact hypergroups.Richard Vrem studied representation theory of compact hypergroups [23]. He showed that, similar to the group case, for every irreducible representation π of a compact hypergroup H, π is of a finite dimension d π . Here we use H to denote the maximal set of all irreducible representations of H which are pairwise inequivalent. The set H equipped with the discrete topology is called the dual space of H.Vrem showed that coefficient functions on compact hypergroups satisfy a hypergroup analogue of Peter-Weyl relation which is as follows [23]. For each pair π, σ ∈ H there exists a constant k π such that for every coefficient functions π i,j and σ k,l , arXiv:1505.01409v3 [math.RT]

Amenability notions of hypergroups and some applications to locally compact groups

“…Proof It is known that as a strong hypergroup satisfies ( P 2 ), (look at [, Proposition 2.4]). Since is isometrically Banach algebra isomorphic to the hypergroup algebra of , one can apply Theorem to the hypergroup to finish the proof.…”

“…Proof. It is known that Irr( ) as a strong hypergroup satisfies ( 2 ), (look at [1,Proposition 2.4] Let be a compact group such that → ∞. Then is called a tall group.…”

“…1 ⊆ W and K * V (1) ∩H \W = ∅ since (x * y)∩(H \W ) = ∅ for all x ∈ K and y ∈ V (1) . Now let us replace K by the compact set K * V (1) .…”

Hardy-Littlewood inequality and $L^p$-$L^q$ Fourier multipliers on compact hypergroups

“…Every compact group is a trivial example of a compact hypergroup. Other essential and non-trivial examples are double coset hypergroups G//H asing from a Gelfand pair (G, H) for a compact group G and a closed subgroup H [27], conjugacy classes of compact Lie groups [46,11], countable compact hypergroups [16,11], Jacobi hypergroups [19,11], hypergroup joins [47] of compact hypergroups by finite hypergroups [6,11].…”

“…6 to get inequality(46) above. The condition(34) for β = 3 by choosing the sequence {µ χn } n∈N := {(1 − a)a −n } n∈N with µ χ 0 = 1 turns out to ben∈N 0 − a) 2 a −2n (1 − a) 3 a −3n = 1 + 1 1 − a n∈N a n = 1 − a + a 2 (1 − a) 2 = (1 − a) 2 − a (1 − a) 2 ,which is finite.…”