Derivations And Cohomological Groups Of Banach Algebras

الأصل · EN

Let B be a Banach A-bimodule and let n≥ 0. We investigate the relationships between some cohomological groups of A, that is, if the topological center of the left module action πℓ:A× B→ B of A⁽²ⁿ⁾ on B⁽²ⁿ⁾ is B⁽²ⁿ⁾ and H¹(A⁽²ⁿ⁺²⁾,B⁽²ⁿ⁺²⁾)=0, then we have H¹(A,B⁽²ⁿ⁾)=0, and we find the relationships between cohomological groups such as H¹(A,B⁽ⁿ⁺²⁾) and H¹(A,B⁽ⁿ⁾), spacial H¹(A,B*) and H¹(A,B⁽²ⁿ⁺¹⁾). 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. Let G be 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)}. We also obtain some conclusions in the Arens regularity of module actions and weak amenability of Banach algebras. We introduce some new concepts as left-weak*-to-weak convergence property [=Lw*wc-property] and right-weak*-to-weak convergence property [=Rw*wc-property] with respect to A and we show that if A* and A**, respectively, have Rw*wc-property and Lw*wc-property and A** is weakly amenable, then A is weakly amenable. We also show to relations between a derivation D:A→ A* and this new concepts.