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- Author or Editor: A. R. Medghalchi x

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*. Johnson has shown that a topological group is amenable if and only if the group algebra

^{8}(G)*L*is amenable. The aim of this research is to define the cohomology on a hypergroup algebra

^{1}(G)*L(K)*and extend the results of

*L*over to

^{1}(G)*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),*

*H*. We prove that

^{1}(A,X^{*})=0*(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 l

_{x}f(µ)=f(d

_{x}*µ). In this case we briefly say that

*L(K)*is left amenable. Johnson also showed that

*L*is amenable if and only if the augmentation ideal

^{1}(G)*I={f? L*

^{1}(G)|∫_{G}f=0}_{0}has abounded right approximate identity. We extend this result to hypergroups.