I think it is fair to say that the fields of Operator Algebras, Operator Theory, and Banach Algebras rely on **Gelfand representation and functional calculus** in a crucial way.



> 
> I am curious about applications of these techniques beyond Functional Analysis. Do you know any?
> 
> 
> 


I am aware of [this thread](https://mathoverflow.net/questions/21090/smooth-gelfand-duality) and [that thread](https://mathoverflow.net/questions/20268/did-gelfands-theory-of-commutative-banach-algebras-influence-algebraic-geometers). I am not primarily interested in interactions of that theory with Algebraic Geometry and Differential Geometry. Also, rather than general ideas, I am looking for concrete examples at the research level. Non research level ones are welcome too as Gelfand's proof of Wiener's lemma, for instance, would be a great fit if I did not know it already.


Thank you.


**Edit:** by Gelfand representation, I refer to [this](https://en.wikipedia.org/wiki/Gelfand_representation#Gelfand_representation_of_a_commutative_Banach_algebra). By functional calculus, I mean either the [holomorphic](https://en.wikipedia.org/wiki/Holomorphic_functional_calculus), or the [continuous](https://planetmath.org/continuousfunctionalcalculus), or the [Borel functional calculus](https://planetmath.org/borelfunctionalcalculus).


**Edit:** in view of the comments below, I will try to say it slightly differently. The tools I mentioned and linked to above belong in Functional Analysis. I am aware of the categorical meaning of Gelfand transform. Out of pure curiosity, I would like to collect results and problems from other areas of Mathematics to which these techniques apply. As Fourier Analysis is such another area even though it has a nontrivial intersection with Functional Analysis, I think Gelfand's proof of Wiener's lemma is a canonical example as it raised the interest in Banach Algebras, as far as I know. The answers by [Asaf](https://mathoverflow.net/a/127999) and [Alain Valette](https://mathoverflow.net/a/128173) are good examples of what I am hoping for.



Gelfand representation and functional calculus applications beyond Functional Analysis
There are some examples in ergodic theory.


Furstenberg's correspondence principle basically relays on Gelfand's representations.


Another example - the functional calculus gives a very easy proof of Von-Neumann's ergodic theorem.


Also, the spectral theorem gives you the existence of spectral measure for a measure preserving system, which itself plays a crucial rule in some of the work in the field.


P.S. in recent years there have been studies which involve ergodic theory and operators on Von-Neumann algebras (and also C\* algebras), those studies probably deal with more FA than the usual ergodic theoretical arguments, but I know next to nothing about those results.


