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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.

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I don't understand why you home in on functional calculus (narrow) and want applications outside functional analysis (incredibly broad). Functional calculus is a tool for understanding the internal structure of various species of Banach algebras. Therefore any sufficiently advanced application of Banach or Cstar algebras probably rests in some way on functional calculus –  Yemon Choi Apr 18 '13 at 4:10
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@Yemon Choi Yes, I mean the Gelfand representation, like here en.wikipedia.org/wiki/…. And not the GNS construction. I might be wrong, but I see the latter as a significantly distinct technique. –  1015 Apr 19 '13 at 0:49
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I interpreted the question "There is a dictionary between commutative Banach algebras and Topology via the Gelfand transform/functional calculus". For me the GNS construction is one line in this dictionary. From Yemon Choi's profile and answers/comments elsewhere on this page, I conclude that he is an expert in this theory, I am not. Sorry for confusing things. –  plusepsilon.de Apr 19 '13 at 9:48
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@Marc Palm. Don't be sorry. Your answers are interesting anyway. –  1015 Apr 19 '13 at 11:00
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The Gelfand transform is an anti-equivalence between the category of commutative unital C*-algebras and the category of compact Hausdorff spaces. Thus I find it strange to exclude functional analysis (to which the category of unital C*-algebras belongs) and geometry (to which the category of compact Hausdorff spaces belongs). So I'm a little confused about what you're hoping for. –  Paul Siegel Apr 20 '13 at 11:55
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7 Answers

I personally like that the Stone–Čech compactification of a Hausdorff Tychonoff space $X$ can be constructed by taking the spectrum of the $\Bbb C$-valued bounded continuous function $C_b(X)$. It is an elegant way to think about a difficult object.

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That's a good one, indeed: the commutative case of the multiplier algebra of a $C^*$-algebra. –  1015 Apr 20 '13 at 16:43
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Gelfand pairs, see

http://en.wikipedia.org/wiki/Gelfand_pair

The fact that you have a subalgebra which is abelian for a non-trivial reason, for which you can compute the Gelfand transform, plays an important role in representation theory, e.g. for semi-simple Lie groups.

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Thank you, it was time for me to realize that. –  1015 Apr 20 '13 at 16:49
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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.

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Thank you. I was not aware of the functional calculus proof of von Neumann's ergodic theorem: this is nice. I confess I did not know the Furstenberg's correspondence principle...Maybe I did not google well enough, but I couldn't see how this relies on Gelfand representation. Could you point me in the right direction? –  1015 Apr 19 '13 at 1:06
    
The basic idea is how to define a proper "measure" on $\mathbb{N}$ which will do the correspondence? As $\mathbb{N}$ does not have any proper "topology" or "measure" it self which will make it an interesting measure space. So Furstenberg looks at the function space $LUC(G)$, which is a $C^{*}$-algebra, and uses the Gelfand representation to construct a measure. Sometimes people avoid the explicit connection to the Gelfand representation by just working with Folner sequences, but if you work all the equivalent definitions for amenability you will find the connection as I stated above. –  Asaf Apr 19 '13 at 6:32
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One more anecdote, I recall that Furstenberg told me once he had given a lecture at Rutgers about Szemeredi's theorem and his correspondence principle, after the lecture Israel Gelfand approached him and said it was very nice, but how he constructed the correspondence, so Furstenberg replied - using Gelfand representation. –  Asaf Apr 19 '13 at 6:34
    
@Asaf Thank you for the sketch of the construction. And for the anecdote! –  1015 Apr 19 '13 at 10:56
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If I understand the question correctly, then the celebrated Corona problem must be such an example. It can be formulated as a classical interpolation problem for bounded, holomorphic functions on the disc but is equivalent to the fact the the open unit disc is dense in the spectrum of the Banach algebra $H^\infty$. The answer is, of course, yes (a famous deep result of Carleson) but the corresponding problem for more complicated domains in the plane or in higher dimensions is still an active area of research with some partial results available.

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That's a good example. Thank you. Apart from the spectrum interpretation of the result, do Carleson's proof, or Wolff's proof, actually use the Gelfand transform? –  1015 Apr 22 '13 at 15:56
    
My understanding of the proof is that Carleson demonstrated the validity of the interpolation property with hard analysis (which, by the way, led to the important concept of Carleson measures). The Banach algebra bit, i.e., the proof of the equivalence of this with the denseness (which is apparently due to Newman), uses the fact that the Gelfand-Neumark transform allows one to consider bounded analytic functions on the open disc and their absolute values as continuous functions on a compactum, the point being that such functions attain their infimum. –  jbc Apr 23 '13 at 13:24
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In complex analysis one can study the polynomial hull $\hat{K}$ of a compact set $K \subset \mathbb{C}^n$, defined as the set of all $x \in \mathbb{C}^n$ such that $|p(x)| \leq \sup_K|p|$ for all polynomials $p$. It is known that there is a canonical homeomorphism between $\hat{K}$ and the Gelfand spectrum of the algebra $A(K)$, the uniform closure of polynomials in $\mathcal{C}(K)$ (see e.g. MR1482798 Alexander, Herbert; Wermer, John Several complex variables and Banach algebras. Third edition. Graduate Texts in Mathematics, 35. Springer-Verlag, New York, 1998. xii+253 pp. ISBN: 0-387-98253-1).

In the paper MR2253160 (2007f:32012) Harvey, F. Reese(1-RICE); Lawson, H. Blaine, Jr.(1-SUNYS) Projective hulls and the projective Gelfand transform. (English summary) Asian J. Math. 10 (2006), no. 3, 607–646, the authors introduce the projective hull of a subset of the complex projective space (using sections of successive powers of the hyperplane bundle, which can be identified with homogeneous polynomials) and develop a theory analogous to that of polynomial hulls. As their tool, for any Banach graded algebra $A_∗$ they construct a topological space $\chi_{A_∗}$, called the projective spectrum of $A_∗$, as the space of continuous homomorphisms $A_* \to \mathbb{C}[t]$ divided by the $\mathbb{C}^\times$ -action corresponding to $\mbox{Aut} (\mathbb{C}[t])$. The space $\chi_{A_∗}$ carries a hermitian line bundle $\lambda$ and there is a natural embedding $A_* \to \oplus_{k \geq 0}\Gamma (\chi_{A_*},\mathcal{O}(\lambda^k))$, called the projective Gelfand transform.

( The notion of) a Banach graded algebra is defined as a commutative $\mathbb{Z}^+$-graded normed algebra $A_*=\oplus_{k \geq 0}A_k$, where each $A_k$ is a Banach space. An example (one which is studied in further detail by Harvey and Lawson) is: $A_k =\Gamma_{hol}(X,\mathcal{O}(L^k))$ with the supremum norm, where $L$ is a holomorphic hermitian line bundle on a complex manifold $X$.

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Thank you these examples and the references. –  1015 Apr 22 '13 at 2:29
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Fourier analysis, i.e., representation theory of abelian groups, has a nice interpretation in terms of the Gelfand transform. The functional calculus is given as convolution operators on a LCA group get sent to multiplication operators on the Pontryagin dual.

Also the representation theory of other groups, such as Lie groups or p-adic groups has profited from Gelfand theory. At the very minimum, it has been a good guiding principle. People rather work with smooth functions in this context. The focus has been here on more explicit description, quantitative analysis and interpretation of C star methods (topology) and von Neumann algebras (measure theory), which are identical in some sense for the type 1 groups. So to say, the functional analysis provides the existence of the measure. Plancherel measure on a type 1 group comes e.g. von a decomposition of right regular rep into irreducibles (vNa decomposition into factors), similarly the spectral side of the Selberg's and Arthur's trace formula. These trace formulas can also be regarded as results in differential geometry/number theory.

A direct application of the GNS-construction/ a concrete example is e.g. the Gelfand-Raikov theorem. Quote from Terry Tao's blog:

"Nevertheless, in the important case of locally compact groups, it is still the case that there are “enough” irreducible unitary representations to recover a significant portion of the above theory. The fundamental theorem here is the Gelfand-Raikov theorem, which asserts that given any non-trivial group element $g$ in a locally compact group, there exists a irreducible unitary representation (possibly infinite-dimensional) on which $g$ acts non-trivially. Very roughly speaking, this theorem is first proven by observing that $g$ acts non-trivially on the regular representation, which (by the Gelfand-Naimark-Segal (GNS) construction) gives a state on the *-algebra of measures on $G$ that distinguishes the Dirac mass $\delta_g$ at from the Dirac mass $\delta_0$ from the origin. Applying the Krein-Milman theorem, one then finds an extreme state with this property; applying the GNS construction, one then obtains the desired irreducible representation."

http://terrytao.wordpress.com/2011/01/23/the-peter-weyl-theorem-and-non-abelian-fourier-analysis-on-compact-groups/

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Where is the functional calculus? Are you referring to Dixmier's results for nilpotent groups? –  Yemon Choi Apr 18 '13 at 4:41
    
Convolution operators on a LCA group $A$ are transformed into multiplication operators on its Pontryagin dual. –  plusepsilon.de Apr 18 '13 at 6:15
    
I am not aware of Dixmier's result. –  plusepsilon.de Apr 18 '13 at 6:19
    
Hmm. I usually think of functional calculus as a way of defining $f(a)$ when $a$ belongs to the algebra, this is sometimes done via Gelfand(+Neumark) theorem but not always. This seems different from "viewing convolution operators as multiplication operators" but perhaps your interpretation of the OP is the correct one! –  Yemon Choi Apr 18 '13 at 6:34
    
I had in mind ams.org/mathscinet-getitem?mr=136684 Opérateurs de rang fini dans les représentations unitaires - perhaps I will post this as a separate answer if I have time tomorrow –  Yemon Choi Apr 18 '13 at 6:38
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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.

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