TITLE: Gelfand representation and functional calculus applications beyond Functional Analysis
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: Very late answer but here is one small application of holomorphic functional calculus in time series/discrete-time stochastic process. Very basic stuff but maybe something a pure mathematician doesn't see everyday.
A baseline time series statistical model is the ARMA (Auto-Regressive Moving-Average) model, which is described by stochastic difference equation
$$
x_t = \sum_{i = 1}^p  \phi_i x_{t-i} + \sum_{j = 0}^q \psi_j \epsilon_{t-j}, 
$$
where $\{ \epsilon_t \}$ is a white noise process, a sequence of independent identically distributed random variables. The statistician is interested in finding a solution of the form 
$$
x_t =  \sum_{i = 0}^{\infty} \theta_i \epsilon_{t-i}.
$$ 
Such a solution necessarily lies in the Banach space $X$ of sequences of bounded (in the $L^2$-norm) random variables. So we are trying to solve the equation
$$
(I - \sum_{i = 1}^p \phi_i B^i) x_t = (\sum_{j = 0}^q \psi_j B^j) \epsilon_t = 
\psi(B)\epsilon_t, 
$$
in $X$, where $B$ is the backward-shift on $X$. If the polynomial $1 - \sum_{i = 1}^p \phi_i z^i$ doesn't vanish on the unit disk, then it has a holomorphic inverse $\theta$ and the desired solution is
$$
x_t = \theta(B)  \psi(B) \epsilon_t. 
$$
The causality/no roots in unit disk condition is always assumed in selecting the model and usually the first thing one checks after fitting data.