I would like to learn Abstract Harmonic Analysis up to a level where research is feasible.
My plan is the following:
- Read Loomis' book completely again. (I already read all the topics related to Banach Algebras.)
- Read Bachman's Book on the subject.
- Finally study Hewitt's comprehensive book on Abstract Harmonic Analysis.
My interest focuses mainly on Banach Algebras and everything to do with $L^1$ and transforms. I am aware that non-commutative topics are very popular today but I would want to focus on Commutative Harmonic Analysis. Is there any paper or book I should definitely read, or do you have any suggestions on how to continue my study afterwards?
I would be very thankful if you could provide a list of the most interesting papers in the subject (and maybe papers which are not that important but indeed very enlightening).
Thanks in advance.
Rudin's 'Fourier analysis on groups' is one of the most enjoyable books I ever read. How well versed are you in classical harmonic analysis? the books by Katznelson and by Helson go a long way toward motivating the general theory. On the abstract side, Arveson's 'invitation to C^* algebras' is recommended. For the interactions of harmonic analysis and probability, Kahane's "Some random series of functions" is a truly wonderful introduction.