Finite dimensional invariant subspaces for algebras of linear operators and amenable Banach algebras

We study a finite dimensional invariant subspace property similar to Fan's Theorem on semigroups for arbitrary Banach algebras AA in terms of amenability of X(A,ϕ)X(A,ϕ), the closed subalgebra of AA generated by the set of all maximal elements in AA with respect to a character ϕ  . As a consequence, we offer some applications to the measure algebra M(G)M(G) and the generalized Fourier algebra Ap(G)Ap(G) of a locally compact group G.