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Paper IPM / M / 2297  


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 H^{1}(A, X^{*})=0 for every Adual 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^{∞}(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 twosided Amodule X with
ax=h(a)x(a ∈ A,x ∈ 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(μ)=f(δ_{x} *μ). 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 ∈ L^{1}(G)∫_{G}f=0} has a bounded right
approximate identity. We extend this result to hypergroups.
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