Multiplication operators

Considering the properties of lattice homomorphisms between the spaces C(K) of all continuous functions on a compact Hausdorff space K, it is given their relation to algebraic homomorphisms. The basic characterization of lattice and algebraic homomorphisms between C(K)-spaces as well as some extension properties of lattice homomorphisms are presented. Multiplication operators on C(K)-spaces are discussed. An abstract M-space (AM-space) with unit as a Banach lattice has an f-algebra structure which is very important. Utilizing the ring structure of an AM-space, some useful ?local approximation? properties of operators on Banach lattices are derived. For example, if we take an AM-space E with unit, then the order projections of the second dual E'' can be approximated ?locally? by the multiplication operators of E. Keywords: Multiplication operator, lattice homomorphism, algebraic homomorphism, Absract M-space (AM-space), orthomorphism, composition operator.