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.
- 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).
- 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?
- 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!