Timeline for Connection on a Hilbert bundle
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Jul 9, 2021 at 21:34 | comment | added | Dmitri Pavlov | @AndréHenriques: A map T⨯T→I(H) is smooth if the associated map F: T⨯T→Hom(R,U(H)) landing in one-parameter groups of unitaries is smooth, where smoothness can be defined by passing to the adjoint map T⨯T⨯R→U(H). For linearity in the second argument, we require that for any t, a, b, the one-parameter groups F(t,a), F(t,b), and F(t,a+b) are related by the Trotter product formula. | |
| Jul 9, 2021 at 21:25 | comment | added | André Henriques | @DmitriPavlov. Given that $I(H)$ is not a vector space, what do you mean by a linear map $T \to I(H)$? Also, what does it mean for a map $T\to Hom(T,I(H))$ to be smooth? | |
| Jul 7, 2021 at 18:58 | comment | added | Hasib | @Dmitry Pavlov: My bad, got it. Thanks a ton. | |
| Jul 7, 2021 at 18:47 | comment | added | Dmitri Pavlov | @Hasib: Why would they go under the rug? The Plancherel measure is still there if you need it. Recall that π_0(R_G) is precisely the unitary dual of G. It has a smooth structure inherited from R_G, and also a compatible measurable structure with a canonical Plancherel measure. | |
| Jul 6, 2021 at 7:09 | vote | accept | Hasib | ||
| Jul 6, 2021 at 7:08 | comment | added | Hasib | @DmitriPavlov: OK, one final question: In the original problem, the family of the Hilbert spaces was indexed by the unitary dual of the Lie group and as a result the inner product structure of the space of sections of the Hilbert bundle was captured using Plancherel measure on it. Do these features go totally under the rug when you're applying the stacks in groupoid set-up to construct connection on the Hilbert bundle? | |
| Jul 5, 2021 at 18:38 | comment | added | Dmitri Pavlov | @Hasib: No, it's T→Hom(T,I(H)). The first T gives a point in T, the second T gives a tangent vector at that point. A connection 1-form sends a point with a tangent vector to an element of the Lie algebra of U(H), i.e., I(H). | |
| Jul 5, 2021 at 17:23 | comment | added | Hasib | @DmitriPavlov: When you're writing the objects of $B_{\nabla}(T)$ as pairs $(H,\nabla)$ do you mean $\nabla: T\rightarrow\hbox{Hom}(\mathcal{G},I(H))$? It occurred to me that it's the Lie algebra $\mathcal{G}$ you wanted to represent as unbounded self-adjoint operators on the Hilbert space $H$. | |
| Jul 5, 2021 at 3:35 | comment | added | David Roberts♦ | @DmitriPavlov ah, excellent, thanks! | |
| Jul 5, 2021 at 3:30 | comment | added | Dmitri Pavlov | @DavidRoberts: The Maurer–Cartan form on U(H) can be defined directly: to a smooth plot P: T→U(H) we must assign a differential 1-form on T, defined as follows: given a point p∈T with a tangent vector q, consider some curve c: R→T such that c(0)=p, c'(0)=q, and take the derivative of P∘c at 0. | |
| Jul 5, 2021 at 3:20 | comment | added | Dmitri Pavlov | @DavidRoberts: Ultraweak topology is the default topology for von Neumann algebras, it is the weak-* topology induced by the predual. For instance, morphisms of von Neumann algebras are defined as ultraweakly continuous *-homomorphisms, or, equivalently, *-homomorphisms that admit a predual. | |
| Jul 4, 2021 at 21:14 | comment | added | David Roberts♦ | I'm curious to know why the choice of ultraweak topology. Also, do you have a reference for why that gives a Lie group, so as to get the Maurer–Cartan form? | |
| Jul 4, 2021 at 20:23 | comment | added | Dmitri Pavlov | @YemonChoi: And points in $R_G$, i.e., isomorphism classes of objects in $R_G({\bf R}^0})$ are precisely points in the unitary dual. | |
| Jul 4, 2021 at 20:22 | comment | added | Dmitri Pavlov | @YemonChoi: Fixing $H$ for simplicity and computing any matrix coefficient gives you a smooth function $R_G→{\bf R}$, which establishes a connection to the Fell topology. (Details need to be checked, of course.) | |
| Jul 4, 2021 at 20:17 | comment | added | Yemon Choi | On further thought, I see that I have a more basic misunderstanding: it is $R_G$ which is the unitary dual, not any particular $R_G(T)$. But could you include in your answer some indication of how one comes down from the stack $R_G$ to what harmonic analysts usually understand by the unitary dual of a locally compact group? | |
| Jul 4, 2021 at 19:59 | comment | added | Yemon Choi | A naive and basic question: in the setting of the OP's question, what is the space T? How is its smooth structure related to the Fell topology on the usual unitary dual of G? | |
| Jul 4, 2021 at 19:51 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |