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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.

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  • $\begingroup$ 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, 2019 at 16:00
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    $\begingroup$ 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, 2019 at 16:00
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    $\begingroup$ 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, 2019 at 16:11
  • $\begingroup$ "Abstract ergodic theorems and weak almost periodic functions" by Eberlein is an excelent resource IMO. $\endgroup$
    – Nick S
    Jun 24, 2020 at 17:56

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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.

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    $\begingroup$ 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, 2019 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, 2019 at 19:17

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