Given a locally compact space $X$, and a set of functions $A\subset C_b(X)$ (where $C_b(X)$ is the set of continuous bounded complex functions)- when can we say that the weak topology generated by $A$ on $X$ is Baire, locally compact or compact?
Does assuming that A is an algebra, C*-algebra etc help?
Edit: I have been unable to find any references on general weak topologies, so any reference would help as well.
Thanks!