Timeline for Integrating a family of vector spaces
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 21, 2018 at 23:12 | comment | added | Mike Shulman | @DanielLitt But unless I misunderstand, it's only as an $L^\infty(X)$-module that the integral has the correct dimension. | |
| Jan 21, 2018 at 20:46 | comment | added | Daniel Litt | @MikeShulman: Suppose I have a sheaf $V$ over a space $X$. Then surely the analogous operation is to take $\Gamma(X, V)=\pi_*V$, where $\pi: X\to *$ is the constant map? This does have the structure of a $\Gamma(X, O_X)$-module, where $O_X$ is the ring of your favorite kind of functions, but one can simply forget this to obtain an object in $Sh(*)$. So surely the correct thing to do here is to take Uri's operation and forget the module structure, no? | |
| Jan 21, 2018 at 20:35 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
minor typos
|
| Jan 12, 2018 at 9:03 | history | edited | Uri Bader | CC BY-SA 3.0 |
deleted 114 characters in body
|
| Sep 28, 2017 at 5:47 | comment | added | Uri Bader | @MichaelBächtold, I am not aware of such an interpretation. Maybe the place to look for such is in the realm of TQFT (interpenetrating the difference of the boundary value as a $V_a^*\otimes V_b$ or as an operator between these spaces). | |
| Sep 27, 2017 at 7:57 | comment | added | Michael Bächtold | Out of curiosity: suppose the base $X$ and the bundle $\pi:V\to X$ are smooth, and the measure is given by a volume form and orientation on $X$, is there also a well known interpretation for $\frac{dV_x}{dx}$ which would make the fundamental theorem of calculus hold when $X$ is an interval? | |
| Sep 26, 2017 at 11:03 | history | edited | Uri Bader | CC BY-SA 3.0 |
deleted 2 characters in body
|
| Sep 26, 2017 at 10:48 | history | edited | Uri Bader | CC BY-SA 3.0 |
added 3252 characters in body
|
| Sep 25, 2017 at 16:35 | comment | added | Mike Shulman | This is really interesting, and I look forward to the updated answer too. I'm not sure whether it will do what I want, though, because the category that the object lives in ($L^\infty(X)$-modules) is dependent on $X$, whereas I would hope/expect an "integral over $X$" to eliminate any dependence on $X$. In particular, I want to consider maps between the integrals $\int V_x \;dx$ over different spaces $X$. But maybe I can put these modules overy varying algebras together into one category somehow. | |
| Sep 25, 2017 at 16:16 | comment | added | Tobias Fritz | Sure, you can equip them with inner products, but then the direct integral will depend on that choice through square-integrability. Anyway, I'm looking forward to an updated answer! | |
| Sep 25, 2017 at 16:12 | comment | added | Uri Bader | In any case, regarding finite dimensional vector spaces, measurability of the family also implies that you can impose on it a measurable inner product, so (even if this is not too natural) you could regard it as a family of Hilbert spaces. But let me repeat: this is not necessary, as direct integral of measurable family of f.d vector spaces could be easily defined. I will update my answer with either a reference or an explanation of the construction in the near future. | |
| Sep 25, 2017 at 16:12 | comment | added | Uri Bader | @TobiasFritz, thanks for the comments. One can define a suitable notion of direct integral of Banach spaces and even more general objects. I am not sure what is the right source for this (maybe Takesaki?), but this is fairly common. | |
| Sep 25, 2017 at 15:51 | comment | added | Tobias Fritz | One can certainly define $\int V_x dx$ by simply omitting any type of integrability condition and this results indeed in an $L^\infty(X)$-module. But unlike in the Hilbert space case, there's no reason to expect it to be finitely generated, which means that von Neumann dimension is generally not applicable. But perhaps one can fix this by considering it as a module over the algebra of all integrable function up to a.e. equivalence (while $L^\infty(X)$ only contains essentially bounded ones). | |
| Sep 25, 2017 at 15:29 | comment | added | Tobias Fritz | Dixmier (in A.73) only defines direct integrals for measurable fields of Hilbert spaces. The Hilbert space structure here is important in that it defines the relevant square-integrability condition. But as the OP has already mentioned in the comments, his vector spaces don't come equipped with an inner product. I've never seen a notion of direct integral for plain vector spaces without any additional structure. What would be a reference for that? (Of course it may well exist and may even be the simpler concept, but it doesn't seem to be covered by the usual notion of direct integral.) | |
| Sep 25, 2017 at 14:04 | history | answered | Uri Bader | CC BY-SA 3.0 |