In this thesis, for a Banach algebra A , we investigate weak amenability of A with respect to a character . We give some necessary conditions for the weak amenability of A with respect to a character and describe a class of Banach algebras that are not weakly amenable with respect to characters . Finally , we give examples of Banach algebras which are weakly amenable with respect to characters but neither weakly amenable nor amenable with respect to characters . Also, we study a finite dimensional invariant suace property similar to Fan's Theorem on semigroups for arbitrary Banach algebras A in terms of amenability of X(A,?) , the closed subalgebra of A generated by the set of all maximal elements in A with respect to a character ? . As a consequence , we offer some applications to the easure algebra M(G) and the generalized Fourier algebra A_p(G) of a locally compact group G.
