Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong sense:

Ad-hoc definition: A bounded continuous function $f\in C_b(G)$ is (right-)$H$-periodic, if $f(gh)=f(g)$ for all $g\in G$.

In particular, if $\pi: G\to G/H$ is the quotient map, then this set of functions is precisely the image of the map $C(G/H)\to C_b(G)$ given by $f\mapsto f\circ\pi$.

My first question: If a function $f\in C_b(G)$ satisfies the $H$-periodicity condition approximately on a large compact set, can it then be perturbed to an $H$-periodic function on a large compact set?

More precisely, does the following hold?

For every $\varepsilon>0$ and compact set $K\subset G$, there exist $\delta>0$ and compact sets $K_1\subset G$, $K_2\subset H$ so that the following holds: If $f_1\in C_b(G)$ is a function with $\|f_1\|_\infty\leq 1$ and $$\max_{g\in K_1}\max_{h\in K_2}\|f_1(gh)-f_1(g)\|\leq\delta,$$ then there exists an $H$-periodic function $f_2$ with $\|f_2-f_1\|_{\infty, K}\leq\varepsilon$.

I have tried to find something like this in the literature, but to no avail.

I figure that if it is true, it should be some clever argument involving only elementary topology and some group theory. In such a case, I really hope for a positive answer in greater generality. Given some C*-algebra $A$, $H$-periodicity makes perfect sense for bounded continuous functions from $G$ to $A$.

My main question: Given a C*-algebra $A$, can an approximately $H$-periodic function in $C_b(G,A)$ be perturbed to an $H$-periodic function in the above sense?

Thoughts: In many cases, this is fairly clear. If $G$ is discrete, then one can just pick a finite representing set of $G/H$, restrict the function $f_1$ there and get $f_2$ by translating this set around with $H$. In cases like $G=\mathbb{R}$ or $\mathbb{R}^n$, it is also (fairly) clear upon choosing a nice set of representatives for the quotient and perturb on its "boundary". For instance, when $G=\mathbb{R}$, then everything reduces to $H=\mathbb{Z}$, and in this case, a function only has to be perturbed near the endpoints of the interval $[0,1]$ to make the endpoints have the same value, and thus one gets an honest $\mathbb{Z}$-periodic function near the original one. However, I have difficulty figuring out what to do in the general, abstract setup.

Assumptions that I am willing to add, but are probably red herrings: $G$ second-countable, $H$ normal, $A$ separable.

  • $\begingroup$ An argument involving only general topology and group theory? You don't want measure theory and integration, convolutions, etc? these are usually useful tools in this context... $\endgroup$
    – YCor
    Feb 11 '16 at 13:13
  • $\begingroup$ Maybe that phrasing was a bit unfortunate. When one faces a question concerning locally compact groups, the mathematical tools you mention are of course natural parts of the theory and important. In particular, when I wrote "group theory", I did not mean it in the purely algebraic sense. $\endgroup$ Feb 11 '16 at 22:38

The answer turns out to be yes. This is Lemma 2.8 from here.

  • $\begingroup$ Reference info: G. Szabo, Rokhlin dimension: absorption of model actions, arxiv.org/abs/1804.04411v1 (Apr. 2018) $\endgroup$
    – YCor
    Jun 16 '18 at 20:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.