Suppose ϕ : S → T \phi :S \to T is an irreducible map of compact Hausdorff spaces, and μ : T → M ( S ) \mu :T \to M(S) the integral representation of an averaging operator for ϕ \phi . We obtain an inequality of the form ‖ μ ( t ) ‖ ≦ ‖ μ ‖ − a ( t ) \left \| {\mu (t)} \right \| \leqq \left \| \mu \right \| - a(t) , where a ( t ) is a(t){\text {is}} a positive number depending on t. From this, some results of Amir and Isbell-Semadeni on P λ {P_\lambda } spaces are shown to follow quickly and a theorem on the isomorphism of certain continuous function spaces is derived.