5
$\begingroup$

I am currently studying Ergodic Theory from Glasner’s book - in it, weakly almost periodic functions play a large role, as well as general “means” and unitary representations of groups on Hilbert spaces.

I cannot seem to grasp the motivation or intuition behind these notions. What text would be best for me to get a better feel for these objects?

Thanks in advance.

$\endgroup$
4
  • $\begingroup$ I'm a bit unsure what good the inuition is meant to be without grappling with examples. Is the motivation what you are primarily looking for? $\endgroup$
    – Yemon Choi
    May 19 '19 at 16:00
  • 1
    $\begingroup$ FWIW, I belong to the school of thought that intuition is something that comes after examples and just before or just after formalism is introduced, but I appreciate that this doesn't work for everyone $\endgroup$
    – Yemon Choi
    May 19 '19 at 16:00
  • 1
    $\begingroup$ Possibly Uri Bader's answer to this MO question mathoverflow.net/questions/232610/… holds some pointers towards the kind of thing you are after $\endgroup$
    – Yemon Choi
    May 19 '19 at 16:11
  • $\begingroup$ "Abstract ergodic theorems and weak almost periodic functions" by Eberlein is an excelent resource IMO. $\endgroup$
    – Nick S
    Jun 24 '20 at 17:56
1
$\begingroup$

To me the best intuition for almost periodicity is Weil’s (1940, Chap. VII), nicely exposed in Dixmier (1982, Chap. 16): Any topological group $\mathrm G$ maps to a “universal” (“Bohr”) compact group $b\mathrm G$, through which all morphisms $\varphi$ of $\mathrm G$ to compact groups (or all finite-dimensional representations of $\mathrm G$) factor uniquely:

$\hspace{5cm}$bG
Now almost periodic functions on $\mathrm G$ are just those $f$ that factor through a continuous $\tilde f$ on $b\mathrm G$, and $f$’s mean is just $\tilde f$’s Haar integral. (When $\mathrm G$ is locally compact abelian, $b\mathrm G$ is the Pontryagin dual of $\mathrm G$’s dual-made-discrete.)

Weak almost periodicity is a variant for which Glasner (p. 47) gives some references; one could add the “very substantial survey” of Štern (2005), which mentions applications to ergodic theory.

$\endgroup$
2
  • 3
    $\begingroup$ A word of warning (for others reading): the WAP compactification of a group is usually no longer a group, but a semi-topological semigroup (multiplication is separately continuous). So some of the intuition from the AP (=Bohr) coimpactification is potentially misleading $\endgroup$
    – Yemon Choi
    May 19 '19 at 15:57
  • $\begingroup$ @Yemon Good point! If I understand Glasner (p. 43) correctly, bG $=$ Gelfand space of AP(G) injects into Z $:=$ Gelfand space of WAP(G); the latter’s $mean$ is integration against the injected Haar measure; and we have WAP(G) $=$ AP(G) $\oplus\ker(mean(|\cdot|))$. $\endgroup$ May 19 '19 at 19:17

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.