Topological centers of module actions and cohomological groups of Banach Algebras

الأصل · EN

In this paper, first we study some Arens regularity properties of module actions. Let B be a Banach A-bimodule and let ZℓB**(A**) and Zℓₐ**(B**) be the topological centers of the left module action πℓ: A× B→ B and the right module action πᵣ: B× A→ B, respectively. We investigate some relationships between topological center of A**, Z₁(A**) with respect to the first Arens product and topological centers of module actions ZℓB**(A**) and Zℓₐ**(B**). On the other hand, if A has Mazure property and B** has the left A**-factorization, then Zℓₐ**(B**)=B, and so for a locally compact non-compact group G with compact covering number card(G), we have Zℓₘ₍G₎**(L¹(G)**)= L¹(G) and Zℓₗ₁₍G₎**(M(G)**)= M(G). By using the Arens regularity of module actions, we study some cohomological groups properties of Banach algebra and we extend some propositions from Dales, Ghahramani, Grønbæk and others into general situations and we investigate the relationships between some cohomological groups of Banach algebra A. We obtain some results in Connes-amenability of Banach algebras, and so for every compact group G, we conclude that H¹w*(L∞(G)*,L∞(G)**)=0. Suppose that G is an amenable locally compact group. Then there is a Banach L¹(G)-bimodule such as (L∞(G),.) such that Z¹(L¹(G),L∞(G))={Lf: f∈ L∞(G)} where for every g∈ L¹(G), we have Lf(g)=f.g.