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Chetan Vuppulury (Jul 07 2020 at 10:06):I tried learning functional analysis (Banach spaces, Operator Algebas, C*-algebras) many many times. But I just never manage to get through, but I really want to learn it because I want to learn Noncommutative Geometry. Does anyone know a good approach to these subjects for the categorically-minded?
Arthur Parzygnat (Jul 07 2020 at 10:49):Look up work by Zbigniew Semadeni. I think he has a functional analysis book. I have to run now, but maybe someone can follow up on this.
Simon Burton (Jul 07 2020 at 12:20):For some reason when category theorists say they want to do functional analysis they end up talking about totally convex spaces. (I'm generalizing from n=1 or 2 samples here...)
Chetan Vuppulury (Jul 07 2020 at 13:26):I'll give them a look. I've seen people recommend Helemskii's 'Lectures and Exercises on Functional Analysis'. Do you know if it is good? Apparently he uses the language of categories.
সায়ন্তন রায় (Jul 07 2020 at 13:27):Chetan Vuppulury said:
I tried learning functional analysis (Banach spaces, Operator Algebas, C*-algebras) many many times. But I just never manage to get through, but I really want to learn it because I want to learn Noncommutative Geometry. Does anyone know a good approach to these subjects for the categorically-minded?
I have heard that this book is what you would call "categorically minded". Although I must confess that I have never read the book.
eric brunner (Jul 07 2020 at 14:25):thanks for the reference. Screen-Shot-2020-07-07-at-7.24.17-AM.png
Arthur Parzygnat (Jul 07 2020 at 16:40):Actually, I don't mean Zbigniew's book on Eilenberg-Moore algebras (though that book is exceptionally well-written, and I highly recommend if you want to learn about EM-algebras). I meant this book "Banach spaces of continuous functions", which is actually a functional analysis book, and which I happily stumbled upon one day in my previous university's library. He also has "Selected topics on functional analysis and categories" but I have no clue what this is (notes, thesis, textbook, a paper?).
Min Ro (Jul 08 2020 at 02:23):Personally, Helemskii's book isn't categorical enough for my tastes. But now I'm remembering back in the days when I wanted to write a text on the matter... oh jeez
eric brunner (Jul 08 2020 at 03:11):expand please. the library copy will be available to me this week. thanks in advance!
Chetan Vuppulury (Jul 08 2020 at 04:20):Min Ro said:
Personally, Helemskii's book isn't categorical enough for my tastes. But now I'm remembering back in the days when I wanted to write a text on the matter... oh jeez
Do you have any recommendations or notes (even incomplete)?
Min Ro (Jul 08 2020 at 04:22):It's probably best to read Helemskii for yourself to see whether you like it or not. It's been far too long for me to give anything detailed, but it seemed like there wasn't as much category theory as I would have liked. But it's also possible I didn't give it a fair chance.
Min Ro (Jul 08 2020 at 04:29):Chetan Vuppulury My personal project was to do a lot of basic functional analysis using metric-enriched or Banach-enriched categories. It seems like a pipe dream now, and there's probably a better way of thinking categorically in an analytic setting. Papers such as Hartig's "Riesz representation revisited" was an inspiration for such a project, but I don't know of any sources that will teach functional analysis from a categorical perspective.
eric brunner (Jul 08 2020 at 04:33):Min Ro said:
It's probably best to read Helemskii for yourself to see whether you like it or not. It's been far too long for me to give anything detailed, but it seemed like there wasn't as much category theory as I would have liked. But it's also possible I didn't give it a fair chance.
fair enough. thanks.
eric brunner (Jul 08 2020 at 04:40):Min Ro said:
... Hartig's "Riesz representation revisited" ...
thank you for the reference.
John van de Wetering (Jul 08 2020 at 12:07):Bram Westerbaan's PhD thesis The Category of Von Neumann Algebras contains a complete introduction (with proofs) into C*-algebras and von Neumann algebras. It doesn't deal with Banach algebras, but it is a lot shorter than the often massive introductory books into operator algebras