Cohomology on hypergroup algebras

There are concepts which are related to or can be formulated by homological techniques, such as derivations, multipliers and lifting problems. Moreover, a Banach algebra A is said to be amenable if H-1(A, X*) = 0 for every A-dual module X*. Another concept related to the theory is the concept of amenability in the sense of Johnson. A topological group G is said to be amenable if there is an invariant mean on L-infinity(G). Johnson has shown that a topological group is amenable if and only if the group algebra L-1(G) is amenable. The aim of this research is to define the cohomology on a hypergroup algebra L(K) and extend the results of L-1 (G) over to L(K). At first stage it is viewed that Johnson's theorem is not valid so more. If A is a Banach algebra and h is a multiplicative linear functional on A, then (A, h) is called left amenable if for any Banach two-sided A-module X with ax = h(a)x (a is an element of A, x is an element of X), H-1(A, X*) = 0. We prove that (L(K), h) is left amenable if and only if K is left amenable. Where, the latter means that there is a left invariant mean m on C(K), i.e., m(l(x)f) = m(f), where l(x)f(mu) = f(delta(x) * mu). In this case we briefly say that L(K) is left amenable. Johnson also showed that L1(G) is amenable if and only if the augmentation ideal I-0 = {f is an element of L-1(G) \ integral(G)f = 0} has a bounded right approximate identity. We extend this result to hypergroups.