A Plancherel Theorem On a Noncommutative Hypergroup

. Let G be a locally compact hypergroup and let K be a compact sub-hypergroup of G . ( G, K ) is a Gelfand pair if M c ( G//K ) , the algebra of measures with compact support on the double coset G//K , is commutative for the convolution. In this paper, assuming that ( G, K ) is a Gelfand pair, we deﬁne and study a Fourier transform on G and then establish a Plancherel theorem for the pair ( G, K ) .


Introduction
Hypergroups generalize locally compact groups. They appear when the Banach space of all bounded Radon measures on a locally compact space carries a convolution having all properties of a group convolution apart from the fact that the convolution of two point measures is a probability measure with compact support and not necessarily a point measure. The intention was to unify harmonic analysis on duals of compact groups, double coset spaces G//H (H a compact subgroup of a locally compact group G), and commutative convolution algebras associated with product linearization formulas of special functions. The notion of hypergroup has been sufficiently studied (see for example [2,4,6,7]).
Harmonic analysis and probability theory on commutative hypergroups are well developed meanwhile where many results from group theory remain valid (see [1]). When G is a commutative hypergroup, the convolution algebra M c (G) consisting of measures with compact support on G is commutative.
The typical example of commutative hypergroup is the double coset G//K when G is a locally compact group, K is a compact subgroup of G such that (G, K) is a Gelfand pair. In [4], R. I. Jewett has shown the existence of a positive measure called Plancherel measure on the dual space G of a commutative hypergroup G. When the hypergroup G is not commutative, it is possible to involve a compact sub-hypergroup K of G leading to a commutative subalgebra of M c (G). In fact, if K is a compact sub-hypergroup of a hypergroup G, the pair (G, K) is said to be a Gelfand pair if M c (G//K) the convolution algebra of measures with compact support on G//K is commutative. The notion of Gelfand pairs for hypergroups is well-known (see [3,8,9]). The goal of this paper is to extend Jewett work's by obtaining a Plancherel theorem over Gelfand pair associated with non-commutative hypergroup. In the next section, we give notations and setup useful for the remainder of this paper.
In section 3, we introduce first the notion of K-multiplicative functions and obtain some of their characterizations. Thanks to these results, we establish a one to one correspondence between the space of K-multiplicative functions and the dual space of G. Then, we define a Fourier tranform on M b (G), the algebra of bounded measures on G and on K(G), the algebra of continuous functions on G with compact support. Finally, using the fact that G//K is a commutative hypergroup, we prove that there exists a nonnegative measure (the Plancherel measure) on the dual space of G.

Notations and preliminaries
We use the notations and setup of this section in the rest of the paper without mentioning. Let G be a locally compact space. We denote by: ii) ∀x, y ∈ G, δ x * δ y is a measure of probability with compact support.
(H2) There is a unique element e (called neutral element) in G such that δ x * δ e = δ e * δ x = δ x , ∀x ∈ G.
(H3) There is an involutive homeomorphism: x −→ x from G in G, named involution, such that: (1) H is non empty and closed in G, Let us now consider a hypergroup G provided with a left Haar measure µ G and K a compact subhypergroup of G with a normalized Haar measure ω K . Let us put M µ G (G) the space of measures in We write simply KxK for a double coset and recall that KxK = coset form a partition of G and the quotient topology with respect to the corresponding equivalence relation equips the double cosets space G//K with a locally topology ( [1], page 53). The natural for all x ∈ G and for all k 1 , k 2 ∈ K. We denote by C (G), (resp. K (G)) the space of continuous functions (resp. continuous functions with compact support) which are one defines the function f on G//K by f (KxK) = f (x) ∀x ∈ G. f is well defined and it is continuous on G//K. Conversely, for all continuous function ϕ on G//K, the function f = ϕ • p K ∈ C (G). One has the obvious consequence that the mapping f −→ f sets up a topological isomorphism between the topological vector spaces C (G) and C(G//K) (see [8,9]). So, for any For a measure µ ∈ M(G), one defines µ by µ (f ) = µ(f ) for f ∈ K(G). µ is said to be K−invariant if µ = µ and we denote by M (G) the set of all those measures. Considering these properties, one defines a hypergroup operation on G//K by: [2, p. 12] ). This defines uniquely the convolution (KxK) * (Ky K) on G//K. The involution is defined by:

Plancherel theorem
Let G be a locally compact hypergroup and let K be a compact sub-hypergroup of G. In this section, we assume that (G, K) is a Gelfand pair.

K-multiplicative functions.
Let us put G b the space of continuous, bounded function φ on G such that: G is called the dual space of the hypergroup G. (1) If φ ∈ G, then φ − ∈ G.
(2) Equipped with the topology of uniform convergence on compacta, G is a locally compact Hausdorff space.
(3) In general, G is not a hypergroup.

Definition 3.2.
A complex-valued function χ on G will be called a multiplicative (resp. Kmultiplicative) function if χ is continuous and not identically zero, and has the property that: A multiplicative (resp. K-multiplicative) function on M b (G) is a continuous complex-valued function F not identically zero on M b (G), and has the property that: For χ ∈ C b (G), not identically zero, let put F χ (µ) = G χdµ for µ ∈ M b (G).
ii) F φ is not identically zero on M µ G (G).

Proof.
i) That is clear that F φ is linear and bounded. Let µ, ν ∈ M b (G). We have 2) Let F be a bounded linear K-multiplicative function on M b (G) not identically zero on The uniqueness stems from proposition 3.3.

Fourier transform on M b (G).
Definition 3.6. Let µ ∈ M b (G), the Fourier transform of µ is the map µ : G −→ C defined by: iii) This comes from theorem 3.5 and ( [4], theorem 6.3G) iv) Let φ belongs to G, we have ii) If f ∈ K (G) and g ∈ K(G), then f * g = f g .

Proof.
i) For any f in K(G), we have ii) Let f ∈ K (G) and g ∈ K(G). For φ ∈ G, we have We therefore extend the spherical Fourier transform to all K(G) with f = f for any f ∈ K(G) and to L 1 (G, µ G ) and L 2 (G, µ G ). We have the following result.
Theorem 3.11. There exists a unique nonnegative measure π on G such that The space f : f ∈ K(G) is dense in L 2 ( G, π).
Otherwise, note that f = f for f ∈ K (G). Indeed since f ∈ K (G) then f ∈ K(G//K) and . So f and f belong to As f = f ∀f ∈ K(G) and G unimodular, we deduce that G f (φ) 2 dπ(φ) = G |f (x)| 2 dµ G (x) ∀f ∈ K(G). By the continuity of the Fourier transform and by application of the dominated convergence theorem, we conclude that G |f (x)| 2 dµ G (x) = G f (φ) 2 dπ(φ) for any f belongs to L 1 (G, µ G ) ∩ L 2 (G, µ G ).

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.