Warner (1966), Hewitt and
Ross (1970), Yap (1970), and Yap (1971) extended the so-called Ditkin’s condition
for the group algebra L1(G) of a locally compact abelian group G to the
algebras L1(G) ∩ L2(G), dense subalgebras of L1(G) which are essential Banach
L1(G)-modules, L1(G) ∩ Lp(G)(1 ≦ p < ∞) and Segal algebras respectively.
Chilana and Ross (1978) proved that the algebra L1(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 pointwise 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.
