Spectral Synthesis in Segal Algebras on Hypergroups

Warner (1966), Hewitt and Ross (1970), Yap (1970), and Yap (1971) extended the so-called Ditkin's condition for the group algebra L\G) of a locally compact abelian group G to the algebras L(G) Π L(G), dense subalgebras of L{G) which are essential Banach LHO-modules, LKG) Π L(G)(1 ^ p < co) and Segal algebras respectively. Chilana and Ross (1978) proved that the algebra L^K) satisfies a stronger form of Ditkin's condition at points of the center Z(K) of K, where K is a commutative locally compact hypergroup such that its dual K is also a hypergroup under point wise operations. Topological hypergroups have been defined and studied by Dunkl (1973), Spector (1973), and Jewett (1975) to begin with. In this paper we define Segal algebras on K and prove that they satisfy a stronger form of Ditkin's condition at the points of Z(K). Examples include the analogues of some Segal algebras on groups and their intersections.