# Textbook recommendations: Weakly almost periodic functions

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?

• 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? May 19 '19 at 16:00
• 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 May 19 '19 at 16:00
• Possibly Uri Bader's answer to this MO question mathoverflow.net/questions/232610/… holds some pointers towards the kind of thing you are after May 19 '19 at 16:11
• "Abstract ergodic theorems and weak almost periodic functions" by Eberlein is an excelent resource IMO. Jun 24 '20 at 17:56

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:
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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.)
• @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) $\oplus\ker(mean(|\cdot|))$. May 19 '19 at 19:17