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I just gave a presentation on exotic group $C^*$-algebras and someone asked why these are studied. I could answer that they can be used to construct $C^*$-algebras with certain properties. However, I couldn't answer the follow-up question: "Why do we study $C^*$-algebras?". So, the question is why do we study $C^*$-algebras.

I know of their connection to bounded operators on Hilbert spaces and the Wikipedia article mentions something about quantum field theory, but these do not feel satisfactory. In conclusion, is there other and better motivation to study $C^*$-algebras?

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    $\begingroup$ See What are the applications of operator algebras to other areas. My answer to States in C${}^*$-algebra and their origins in physics might help too. $\endgroup$
    – Nik Weaver
    Apr 19 at 10:39
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    $\begingroup$ I changed the notation, $C^*$ is the usual form, not $\mathbb{C}^*$ $\endgroup$ Apr 19 at 10:58
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    $\begingroup$ Seriously, if you’re interested in the original historical motivation for $C^\ast$-algebras, see Nik Weaver’s answer in his second link. Compare the Kadison–Singer problem, whose original formulation comes straight out of quantum mechanics as formulated in terms of $C^\ast$-algebras of bounded observables. [BTW, the recent solution of the Kadison–Singer problem depends crucially on Weaver’s reformulation of Anderson’s reformulation, if I understand the basic history correctly?] $\endgroup$ Apr 19 at 12:24
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    $\begingroup$ Can I read between the lines, and guess you are a student? May I ask: at what point in your education? It could be (I of course cannot know for sure) that the question, in response to a talk you gave, might have been more asking "Do you, as a student, know something of the wider history of this subject?" rather than (as I think people here might be tempted to read) "Why is this area of Mathematics important?" Just a guess... $\endgroup$ Apr 19 at 15:40
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    $\begingroup$ @MatthewDaws very good point. $\endgroup$
    – Nik Weaver
    Apr 19 at 16:29
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$\DeclareMathOperator{\Spec}{Spec}$I think the original motivation was to construct „Spectral Calculus“: given a continuous function $f$ on the spectrum of an operator $A$, you want to have an operator $f(A)$ such that $\Spec(f(A))=f(\Spec(A))$.

Given $f$, you approximate it by polynomials $p_n(z,\overline{z})$, and then you want $p_n(A,A^*)$ to converge to an operator that will be defined to be $f(A)$. For the latter question you need to understand $C^*$-algebras.

For a normal operator $N$, the Gelfand-Naimark theorem Gelfand duality theorem gives an isomorphism of $C^*$-algebras $C_0(\Spec(N))\simeq E(N,N^*)$ (the latter is the $C^*$-algebra generated by $N$ and $N^*$), which is what you need to make spectral calculus work.

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    $\begingroup$ Why do you think this was the original motivation? $\endgroup$
    – Nik Weaver
    Apr 19 at 11:24
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    $\begingroup$ I shouldn’t have said „original motivation“, because that seems to have been quantum mechanics. But it seems to have been the work of Gelfand which popularized C*-algebras inside mathematics. $\endgroup$
    – ThiKu
    Apr 19 at 14:00
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    $\begingroup$ The Gelfand–Neumark theorem (proved in 1941) does not say what you claim it says. Rather, it proves that any C*-algebra is isomorphic to a norm-closed *-subalgebra of B(H). The result that you cited is known as the Gelfand duality theorem and was proved in 1939 by Gelfand alone, without Neumark. $\endgroup$ Apr 19 at 15:38
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    $\begingroup$ @ThiKu I'm happy to agree that C*-algebras began with the paper of Gelfand and Naimark, but I don't think the motivation you describe is anything like the motivation for that paper. $\endgroup$
    – Nik Weaver
    Apr 19 at 16:27
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    $\begingroup$ @DmitriPavlov Also: having taken a quick look at the MathReview for the German translation of Gelfand's 1939 paper, it doesn't mention any Cstar structure; this appears to be the paper which constructs/defines the Gelfand representation for unital commutative Banach algebras. So perhaps the theorem for commutative Cstar algebras is after all attributable to the later Gelfand-Neumark paper? $\endgroup$
    – Yemon Choi
    Apr 20 at 4:23

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