Suppose we are given a locally compact space $X$ with $C_b(X)$ denoting the continuous bounded complex or real functions on $X$.

Now, if $A\subset C_b(X)$ is given, I am trying to figure out when the weak topology induced by $A$ on $X$, i.e. the weakest topology on $X$ making functions in $A$ continuous, has the Baire property.

In particular, if $A$ separates the points of $X$ and thus induces a Hausdorff topology, what extra properties of $A$ may ensure local compactness, compactness? Does assuming that $A$ is a C*-algebra help?

I haven't been able to find any references on general weak topologies, so I would really appreciate any input or intuition.