Characterizations of the projection bands and some order properties of the lattices of continuous functions

Abstract

We show that for an ideal H in an Archimedean vector lattice F the following conditions are equivalent:

• H is a projection band;

• Any collection of mutually disjoint vectors in H, which is order bounded in F, is order bounded in H;

• H is an infinite meet-distributive element of the lattice \({\mathcal {I}}_{F}\) of all ideals in F in the sense that \(\bigcap \nolimits _{J\in {\mathcal {J}}}\left( H+ J\right) =H+ \bigcap {\mathcal {J}}\), for any \({\mathcal {J}}\subset {\mathcal {I}}_{F}\).

Additionally, we show that if F is uniformly complete and H is a uniformly closed principal ideal, then H is a projection band. In the process we investigate some order properties of lattices of continuous functions on Tychonoff topological spaces.