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Motivation

I got interested in abstract harmonic analysis when I was reading representation theory of groups. In chapter 4 of J.-P. Serre's classic text Linear Representations of Finite Groups, the author explains (in a sketchy manner) how the results on representations of finite groups could be extended to the case of (locally) compact groups. I was surprised to find that this is discussed in the field abstract harmonic analysis.

Questions

  1. Is it possible (or rather, advised) to study abstract harmonic analysis without knowing any harmonic analysis?

I currently know nothing about Fourier/harmonic analysis, except those mentioned in functional analysis (e.g., Fourier expansion on $L^2$). However, I'm not interested in "hard analysis"; I only want to learn about the representation of locally compact groups. It seems that "classical" Fourier analysis is mostly hard analysis (correct me if I'm wrong).

  1. How much should I know about Banach algebras?

I've checked out Folland's text and found that the first chapter is about Banach algebras. I only know the rudiments in this subject, barely enough to prove the spectral theorem for bounded normal operators on a Hilbert space. Should I read a separate book on Banach algebras? If so, what text would you recommend?

  1. What text would you recommend for abstract harmonic analysis?

There are texts by Hewitt & Ross, Loomis, Folland, Deitmar & Echterhoff, etc., and I don't know which one is better...

Thanks for any advice!

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  • $\begingroup$ May be you want to read about topological groups first... I was in similar situation.. (I am no more studying representation theory).. I like Follands book amazon.com/Abstract-Harmonic-Analysis-Advanced-Mathematics/dp/… You can start reading second chapter on Locally compact groups... Once you finish second chapter, you will get to know what to read based on your requirement... You might need to know some measure theory to understand Haar measure and all that... I would be interested to discuss something in chapter $2$ if you have some specific question after reading.. $\endgroup$ – Praphulla Koushik Feb 12 at 6:33
  • $\begingroup$ @PraphullaKoushik Thanks for your advice! In fact I have already read the proof of existence and uniqueness of Haar measure from Folland's real analysis text. I'll definitely check out the text you linked and see if I can handle it. :) $\endgroup$ – Colescu Feb 14 at 3:16
  • $\begingroup$ I do not know about Haar measure being mentioned in Folland's real analysis text... You can definitely handle :) :) $\endgroup$ – Praphulla Koushik Feb 14 at 5:55

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