harmonic analysis?

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Oct 17, 2020 at 0:13	history	edited	Chetan Vuppulury	CC BY-SA 4.0	fixed latex
Jun 22, 2013 at 7:13	comment	added	Eric		@The User: Many of the other answers were not applications of category theory in this sense. The difficulty in both asking and answering a question along these lines was already addressed comments, as well as in some of the answers.
Jun 20, 2013 at 15:58	comment	added	The User		-1 Equivalences and adjoints arise everywhere. But I would not consider it an “application of category theory” if no theorem from category theory is used for it.
Dec 16, 2011 at 0:16	comment	added	Eric		@Yemon: Definitely, reference to the morphisms is necessary to fully realize the categorical viewpoint. My category theorist friend would be horrified to see me make no mention of what the functor does on morphisms! I hope he can forgive me if he ever reads this.
Dec 15, 2011 at 20:19	comment	added	Benjamin Steinberg		One can prove the universal property of the Stone-Čech compactification by provings its associated C^*-algebra has the appropriate property. To do this one should really look at bounded continuous functions and the spectrum as giving an adjunction which restricts to an equivalence on the usual suspects.
Dec 15, 2011 at 18:20	comment	added	Yemon Choi		@Johannes: but does the equivalence of categories (as opposed to the object-level equivalence) really get used in spectral theory? My impression was that knowing the Gelfand transform maps onto C(max ideal space) -- and of course the Gelfand transform belongs to the province of Banach algebras, rather than $C^\ast$ algebras -- is all that's needed for e.g. continuous functional calculus.
Dec 15, 2011 at 12:21	comment	added	Johannes Ebert		+1 because you mentioned it as an application to analysis. Knowing that a commutative C-algebra is the same as functions on a compact space is helpul, e.g. in spectral theory. The other direction is much less useful (knowing that a compact space is the same as the spectrum of a C-algebra does not tell you anything.
Dec 15, 2011 at 4:21	comment	added	Yemon Choi		Without reference to the morphisms, of course, this theorem is in many if not most books on functional analysis. (The force of the categorical viewpoint is the relationship between various $C^\ast$-algebras, not just the fact that the Gelfand transform maps onto C(max ideal space)))
Dec 15, 2011 at 2:13	comment	added	Eric		As far as I understand, this is one of the jumping off points in certain non-commutative geometry/pseudo-geometry programs. When studying non-commutative C*-algebras, the thought is that they should correspond to some non-commutative space analogous to the commutative case.
Dec 15, 2011 at 2:08	history	answered	Eric	CC BY-SA 3.0