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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.

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  • $\begingroup$ I think that the edit to the title is erroneous. $\endgroup$ – Douglas Somerset Jul 8 '19 at 22:38
  • $\begingroup$ @DouglasSomerset It indeed is! $\endgroup$ – Merry Jul 9 '19 at 19:01
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    $\begingroup$ If you assume that $A$ is a C$^*$-algebra then $X/R$ will be the Gelfand space of $A$, where $R$ is the equivalence relation on $X$ consisting of points not separated by $A$. So $X/R$ will be locally compact and Baire, and hence $X$ will be too in the weak topology. $\endgroup$ – Douglas Somerset Jul 9 '19 at 20:42
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    $\begingroup$ I don't think assuming that $A$ is a C* algebra is essential actually, since a collection of functions generates the same topology as the collection of all possible C* combinations of them, and a collection generates the same topology as its closure in $C_b(X)$. $\endgroup$ – erz Jul 11 '19 at 9:15

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