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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
$L^{\infty}(G)$. 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
$ax=h(a)x(a\in A,x\in X), H^1(A, X^*)=0$. 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., $m(l_xf)=m(f)$, where $l_xf(\mu)=f(\delta_x *\mu)$. In this
case we briefly say that $L(K)$ is left amenable. Johnson also
showed that $L^1(G)$ is amenable if and only if the augmentation
ideal $I_0=\{f\in L^1(G)|\int_Gf=0\}$ has a bounded right
approximate identity. We extend this result to hypergroups.
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