Fixed point properties related to character amenable Banach algebras

For a Banach algebra U, we investigate several fixed point properties for U. with respect to a non-zero character phi on U. As the main results, we obtain some fixed point characterizations for phi-amenability of U. We also describe phi-amenability of U in terms of the Hahn-Banach extension property. As a consequence, we offer some applications to the group algebra and the Fourier algebra of a locally compact group.