My setting is the following : let $G$ be a topological group and $X$ be a topological space. I have the head filled with two possible definitions for a continuous action of $G$ on $X$.
The first could be called the separately continuous action and means that every element $g\in G$ acts via a continuous map of $X$ into itself, and that the action $G \rightarrow X, \,\, g\mapsto g \cdot x$ on every element $x\in X$ is continuous. Equivalently, this means that we have a continuous morphism of group from $G$ into $\mathcal{C}^0(X,X)$ where the last group structure is for the composition, which is continuous for the simple convergence topology on $\mathcal{C}^0(X,X)$ (which is not a toplogical group structure in general, but this is not useful).
The second one could be called the joint continuous action. It means a group action of $G$ on $X$ such that the deduced map $G\times X \rightarrow X$ is continuous.
I have seen some conditions for the two definitions to be equivalent (for instance for $X$ with extrastructure in Locally analytic vectors in representations of locally $p$-adic analytic groups by M. Emerton). However, seing the lectures of P. Scholze and D. Clausen and especially the definition of the site $G$–$\mathrm{pfsets}$, I came to have the question in the following specific setting : $X$ (resp. $G$) is a profinite set (resp. group). I have also manage to prove that in the case where both limits defining the profinite things can be taken countable, the two conditions are equivalent. I have tried some generalization (but the lack of Mittag-Leffler systems makes it difficult) and counterexamples without success. So my question is the following :
- Are both definitions equivalent for this profinite setting ?
- Are they more general conditions on the category indexing the limits so that the two conditions are equivalent ?
Best Answer
Both definitions are equivalent in much greater generality. I believe the profinite case has as more direct proof, perhaps it can be found in the book of Ribes and Zalesski.
There is a very general result by Robert Ellis (Ellis, Robert Locally compact transformation groups. Duke Math. J. 24 (1957), 119–125) that covers a much more general situation than yours. If $G$ is a group with a topology for which multiplication is separately continuous in each variable and if the action map $\pi\colon G\times X\to X$ is separately continuous in each variable, and $X$ is locally compact, then $\pi$ is continuous. His paper is written in a slightly hard to digest language and is nontrivial. He views $G$ as a subgroup of the homeomorphism group of $X$ (but the action doesn't really need to be faithful) and expresses the other continuity condition as saying that the topology on $G$ contains the topology of pointwise convergence.