Cohomology on hypergroup algebras

Abstract

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 H1(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 8(G). Johnson has shown that a topological group is amenable if and only if the group algebra L1(G) is amenable. The aim of this research is to define the cohomology on a hypergroup algebra L(K) and extend the results of L1(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? A, x? X),H1(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(lf)=m(f)x, where lxf(µ)=f(dx*µ). 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={f? L1(G)|∫Gf=0}0 has abounded right approximate identity. We extend this result to hypergroups.