8

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?

David Roberts
  • 33,851
Emiel Lanckriet
  • 81
  • 10
    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. – Nik Weaver Apr 19 '21 at 10:39
  • 4
    I changed the notation, $C^$ is the usual form, not $\mathbb{C}^$ – David Roberts Apr 19 '21 at 10:58
  • 3
    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?]

     – Branimir Ćaćić Apr 19 '21 at 12:24
  • 4
    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... – Matthew Daws Apr 19 '21 at 15:40
  • 2
    @MatthewDaws very good point. – Nik Weaver Apr 19 '21 at 16:29
  • I am a student finishing my masters degree, but I have only learned about $C^*$-algebras in the last year. The question came from a fellow student who was also interested in the use of this area of mathematics. It is correct that I don't know a lot about the wider history of the subject, but I would say that the question is more "What did we gain from studying it?", than "Why did we start studying it?" although both are interesting questions. – Emiel Lanckriet Apr 20 '21 at 09:32
  • 1
    Not an answer to "why C-algebras?" but a comment on the more focused "why exotic group C-algebras?": one perfectly reasonable answer is "to build interesting examples of C-algebras". Another is that the K-theory of group C-algebras (and crossed products) is a receptacle for interesting invariants: equivariant indices etc. Different C*-algebras attached to the same group can have different K-theory, and it appears that for some purposes the nicest K-theoretic properties are enjoyed by "exotic" algebras in between the two standard ones. See the work of Baum-Guentner-Willett and others. – user85913 Apr 20 '21 at 12:55
  • 1
    One motivation is quantum statistical mechanics, see e.g. "Operator Algebras and Quantum Statistical Mechanics 1", https://www.springer.com/gp/book/9783540170938 . – jjcale May 19 '21 at 21:31

1 Answers1

1

$\DeclareMathOperator{\Spec}{Spec}$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.

ThiKu
  • 10,266
  • 5
    Why do you think this was the original motivation? – Nik Weaver Apr 19 '21 at 11:24
  • 1
    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. – ThiKu Apr 19 '21 at 14:00
  • I guess this depends whether you consider the history of C-algebras to start with the abstract definition in today’s terms by Gelfand-Naimark 1943, or with the concrete examples that have been studied before (like e.g. von Neumann algebras, coined W-algebras at the time) in physics. – ThiKu Apr 19 '21 at 14:08
  • 2
    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. – Dmitri Pavlov Apr 19 '21 at 15:38
  • 2
    @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. – Nik Weaver Apr 19 '21 at 16:27
  • I agree with @NikWeaver - I would be very surprised if this was the original motivation. What you describe here seems to be a case of reading with hindsight. Just because commutative Cstar algebras provide a nice framework in which to develop the spectral theory of bounded normal operators on Hilbert space, that doesn't mean the latter was the motivation to study general Cstar algebras – Yemon Choi Apr 20 '21 at 04:11
  • 1
    @DmitriPavlov I defer to your knowledge of the original literature, but I feel it should be pointed out (just for people who are comparing different accounts and textbooks) that there are some standard textbooks in functional analysis which use the name "Gelfand-Neumark theorem" for the representation theorem in the commutative case (and at the time it was not known as the duality theorem, because the categorical perspective had not yet been formulated) – Yemon Choi Apr 20 '21 at 04:16
  • 2
    @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? – Yemon Choi Apr 20 '21 at 04:23
  • @YemonChoi: You can look at the Gelfand–Neumark paper, Lemma 1 states Gelfand duality and proves it by referencing Gelfand's 1939 paper. – Dmitri Pavlov Apr 20 '21 at 04:27