Characterization of Best Approximation Points with Lattice Homomorphisms

In this paper we prove some characterization theorems in the theory of best approximation in Banach lattices. We use a new idea for finding the best approximation points in an ideal. We find the distance between an ideal I and an element x by using lattice homomorphisms. We introduce maximal ideals of an AM space and characterize other ideals by the maximal ideals. We also give a new representation for principle ideals in Banach lattices that is a majorizing subspace and we show that these principle ideals are proximinal. The role of lattice homomorphisms in this paper is very important.