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Paper   IPM / M / 2297
School of Mathematics
  Title:   Cohomology on hypergroup algebras same
  Author(s):  A. R. Medghalchi
  Status:   Published
  Journal: Studia Sci. Math. Hungar.
  Vol.:  39
  Year:  2002
  Pages:   297-307
  Supported by:  IPM
  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(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(aA,xX), 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(lxf)=m(f), where lxf(μ)=fx *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 I0={fL1(G)|∫Gf=0} has a bounded right approximate identity. We extend this result to hypergroups.

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