Timeline for Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?
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15 events
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| Oct 6, 2020 at 0:00 | comment | added | David Roberts♦ | @SimonHenry yeah, I thought that very unlikely, but I think it deserved pointing out, even if only to shut down red herrings. | |
| Oct 5, 2020 at 23:55 | comment | added | Simon Henry | @DavidRoberts The fact that they have the same homotopy type is somewhat of an accident. In general $BG$ and $G^*$ do not have the same homotopy type. | |
| Oct 5, 2020 at 23:50 | comment | added | David Roberts♦ | Both of these groupoids are in fact Lie 2-groups, so that would give extra structure to their $C^*$-algebras. Also, it shouldn't escape anyone's notice that $\mathbb{U}$ and $\mathbf{B}\mathbb{Z}$ have the same homotopy type. | |
| Oct 3, 2020 at 18:21 | comment | added | Mirco A. Mannucci | @BenjaminSteinberg I am (alas) an ignoramus, so I need more details. What you say is fascinating to me, especially because the real motivation behind my question (aside my delirium on QM) is the true connection between non commutative geometry and topos. As you know, grothendiek topoi are (up to equivalence) the equivariant sheaves for a localic topos. So we are here dangerously close to that. But what is exactly that Exel algebra? I would appreciate if you expanded this comment into some kind of answer (I do understand that it would be incomplete, but that is ok). | |
| Oct 3, 2020 at 17:48 | comment | added | Benjamin Steinberg | There is a sort of duality between etale groupoids with totally disconnected unit space and boolean inverse semigroups. The Exel C*-algebra of the boolean inverse semigroup is the groupoid C*-algebra. I'm not sure how much further one can take this among groupoids even in the etale case | |
| Oct 3, 2020 at 17:02 | comment | added | Mirco A. Mannucci | extremely interesting. This is not only an intriguing mathematical question: far from it, I think it has great implications on the very foundation of QM. In QM one talks about states and operators (the hilbert space and the operator algebra), but my gut feeling is that this is a fundamentally incomplete approach: EVERYTHING is an operator. Somehow this is related to the fact that, unlike the Gelfand duality for commutative algebras, which is a perfectly clear duality, the GNS construction is not. But I am jumping ahead. Let me process the thing a bit more, and I will come back. Merci bien | |
| Oct 3, 2020 at 16:51 | comment | added | Simon Henry | Absolutely. Convolution algebra have a lot of additional structure. The most documented example are these cartan subalgebra I mentioned in my original answer (for which there are partial result in the spirit of a n.c. duality) . But there are many other kind of structure. For example, the category of representation of a convolution algebra can be described in terms of representation of the groupoids and inherits a monoidal structure because of this. That does not quite corresponds to a bialgebra structure on the $C^*$-algebra, but that's something in this direction. | |
| Oct 3, 2020 at 16:27 | comment | added | Mirco A. Mannucci | @SimonHenry thanks for the additional remarks. I obviously need to do a bit of homework here. So, summing up, you point is: rather than hoping in a duality at the level of C* algebras (no algebraically definable subcategory of C*- Alg would do) you seem to suggest that one should look for a category of structured C* algebras (perhaps algebraic quantum groups, or something along those lines) to search for the infamous non-commutative duality. Am I correct? If yes, then the obvious question is: the convolution algebras one gets from groupoids do carry this additional structure? | |
| Oct 3, 2020 at 16:20 | comment | added | Simon Henry | (...one kind of Quantum groups are some sort of $C^*$-Hopf algebra, and there is a Fourier duality theory for these, there are also purely algebraic notion of Quantum group, but that's not what I was refering too) | |
| Oct 3, 2020 at 16:19 | comment | added | Simon Henry | @MircoA.Mannucci : The reason I talked about fourier transform is because the example generalizes to any locally compact topological group $G$: $C^*(BG) \simeq C(G^*)$ where $G^*$ is the pontryagin dual, and this isomorphism is the Fourier duality for the group $G$. Even for more general groupoids, these kind of isomorphisms often feels like some kind of genealized Fourier transform. I don't think you can define a Fourier dual of a $C^*$-algebra in general, but there is definitely a notion of Fourier dual for $C^*$-algebra with additional structure, see for e.g. the theory of Quantum groups | |
| Oct 3, 2020 at 15:56 | comment | added | Mirco A. Mannucci | Cher Simon, first of all KUDOS: your answer not only deserves a GREEN by MO standards, but a ULTRAGREEN. Why? not full of technicalities, but very high on content: a simple enlightening example, straight to the core. But: you got yourself in trouble: rather than satisfying my appetite, you just wetted it. For instance: you talk about the "fourier transform" of C* algebras: is this an involutive endofunctor of the category? If yes, maybe there is an analogue in a suitable sub-category of groupoids (switching point and arrows) that would (perhaps) explain this oddity... | |
| Oct 3, 2020 at 14:46 | vote | accept | Mirco A. Mannucci | ||
| Oct 3, 2020 at 14:08 | comment | added | Nik Weaver | This is a great answer. | |
| Oct 3, 2020 at 13:24 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Oct 3, 2020 at 13:13 | history | answered | Simon Henry | CC BY-SA 4.0 |