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I am teaching a course on locally compact groups and their representation theory and I am at a point where I would like to introduce continuous representations as generally as possible and provide meaningful examples. The end goal is of course to introduce the left [right] regular representation of $G$ on $L^2(G) = L^2(G, \mu; \mathbb{C})$, where $G$ is locally compact and $\mu$ is Haar measure, but I was hoping to be able to start with other (non-Hilbert) function spaces which might be more natural from a purely topological perspective—for instance, the space $C(G) = C(G; \mathbb{C})$ of continuous complex-valued functions on $G$, equipped with the compact-open topology, and where the action of $G$ is by left (or right) translation. However, I'm not nearly familiar enough with (or sufficiently well-versed in) the compact-open topology to prove that this action is actually continuous—or to discern whether it is only continuous for certain (e.g., locally compact) $G$. I thought I had an argument for compact $G$ but I know realize that it was incomplete. A reference would be greatly appreciated.

Even if the action on $C(G)$ should turn out to not be continuous in general, this would still be instructive to know. My next question would then be if the restricted action on $C_c(G)$, equipped with any of the (possibly very different) topologies considered here, is continuous for $G$ locally compact; it should definitely be true for the topology coming from the inclusion $C_c(G) \hookrightarrow L^2(G)$, but I don't know about the other topologies (provided they are indeed different).

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  • $\begingroup$ Just to clarify: by continuity of a representation of G on a TVS V, I assume you mean "strong operator continuity", i.e. continuity of the orbit maps $g\mapsto gv$ for each $v\in V$? $\endgroup$
    – Yemon Choi
    Jun 27 at 15:34
  • $\begingroup$ @YemonChoi sorry for the delay. By continuity (or strong continuity) I mean that the map $G \times V \to V$, $(g, v) \mapsto g \cdot v$ is jointly continuous. This implies "separate continuity" (i.e., continuity of the orbit maps + continuity of each linear map $v \mapsto g \cdot v$) but as far as I know the converse is only known (Bourbaki's "Integration") if $V$ is locally convex, Hausdorff and barrelled. $\endgroup$
    – epitaph
    Jul 15 at 16:27
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In order to be explicit, let me consider the case where the group is the real line. There are a number of relevant function spaces: bounded continuous functions, continuous functions, continuous functions with compact support, $L^p$-spaces (for Haar measure). These are, respectively, a Banach space, a Fréchet space, an $LF$-space, Banach spaces. For all of these, with two exceptions, the representations are continuous. The two exceptions are those of bounded (continuous or measurable) functions. In those cases, there are as substitutes perfectly respectable complete l.c. topologies (strict topologies) for which they are again continuous.
Analogous statements hold for arbitrary locally compact groups (see one of the plethora of books on abstract harmonic analysis).

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  • $\begingroup$ Since posting the question I have found a proof for the continuity of the representations on $C(G)$, $C_c(G)$ and $L^p(G)$ ($p < \infty$) in Bourbaki, "Integration", Chapter VIII, §2, No.s 3 and 5. None of my go-to reference books on harmonic analysis (Hewitt-Ross, Folland, G. Warner) seem to contain these proofs in full generality—they either restrict to (continuous) Banach or even unitary representations or (in Warner's case) focus on Lie groups and representations on spaces of smooth functions. If you add a concrete reference for the two remaining cases I will accept your answer. $\endgroup$
    – epitaph
    Jul 15 at 16:31

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