It is well known that the set of self-adjoint elements in a $C^*$ algebra naturally forms a poset. I assume that the general properties of this poset are known. Can anybody point me to a reference where the basic properties of that poset are established?
In particular, I'm interested to know when the poset will be a lattice, a complete lattice, and when the joins/meets behave nicely with respect to the $C^*$ algebra operations.