[Math] Gelfand representation and functional calculus applications beyond Functional Analysis

Question

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 and that thread. 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. By functional calculus, I mean either the holomorphic, or the continuous, or the Borel functional calculus.

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 and Alain Valette are good examples of what I am hoping for.

Answer 1

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.