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In this essay we study the concepts of nuclearity in C-algebras, and amenability in Banach algebras. A proof on the "every amenable C-algebra is nuclear" is studied. A Banach algebra A is called amenable if every continous derivation D : A ! E , is inner i.e. there exists anif in this denition E is a dual Banach A-module and norm continuity replace by w-continuity then we have Connes-amenability; Connes-amenability was rst considered for von Neumann algebras in [J-K-R] (and thus should perhaps be called Johnson-Kadison-Ringrose amenability). The reason why this notion of amenability is usually associated with A. Connes are his papers [Conn 1] and [Conn 2] The name "Connes-amenability" seems...