Continuity of left regular representation on space of continuous functions

Question

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).

Answer 1

I think it is important to mention a non-pathological counter-example (although we do expect most such actions to be continuous $G\times V\to V$), namely, the action of $G$ on the space $V$ of all bounded continuous functions on $G$, with sup norm. Already on $G=\mathbb R$, this action is not continuous, because functions need not be uniformly continuous. E.g., $v=\sin(x^2)$ is not acted-upon continuously...

From the other side, the action of $G$ on compactly-supported continuous functions $C^\infty_c(G)$ is continuous, where the latter has the natural strict-colimit ("LF") topology. This reasonably leads to the corollary for any space $V$ of functions in which $C^\infty_c(G)$ is dense (and the inclusion map is continuous).

That basic counter-example is important to remind us that the action of a group of invertible operators on a Hilbert space rarely gives a (jointly) continuous $G\times V\to V$, so integrals-of-operators $\int_G \varphi\cdot g \; dg$ do not converge in the uniform operator norm topology, but only in the strong topology on operators.

Answer 2

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).