Representing orthomorphisms and similar operators as multiplication operators

In the talk, we will review some results on representation of orthomorphisms. An operator $T$ on a vector lattice is called an orthomorphism if it is order bounded and leaves every band (i.e., an order closed order ideal) invariant. This class includes central operators, i.e., operators that are dominated by a scalar multiple of the identity. If we represent our vector lattice as a space of continuous functions, orthomorphisms correspond to multiplication operators. This representation allows one to deduce various properties of orthomorphisms. Furthermore, under some mild assumptions, every disjointness preserving operator may be represented as a weighted composition operator.