Timeline for Unitary representations of Fuchsian and Kleinian groups
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 1 at 7:22 | comment | added | David Gao | Dixmier’s book should work. Takesaki’s Theory of Operator Algebras I also has a section on direct integrals, as well as a chapter on type decomposition for von Neumann algebras. | |
| Dec 1 at 6:51 | comment | added | user82261 | @DavidGao This helps a lot. What’s a good reference for von von Neumann algebras and group representations where I can look up more details? Diximer’s book? | |
| Dec 1 at 5:22 | comment | added | David Gao | (Though, if you just want the representation to be decomposed into type I factors, then that’s equivalent to the vNa generated being type I, or equivalently the commutant being type I. That’s when the direct integral decomposition is of finite-dimensional factors and $B(\ell^2)$. If you want to eliminate $B(\ell^2)$, I guess saying the representation is both of type I and is induced by a trace on the group works. This always happens for virtually abelian groups.) | |
| Dec 1 at 5:07 | comment | added | David Gao | A given representation is a direct integral of irreducible representations iff the commutant is abelian. | |
| Dec 1 at 4:50 | comment | added | user82261 | @DavidGao Is it possible to determine whether a given representation can split into a direct integral of irreducible representations? Any techniques for doing this? The group Gamma is not virtually Abelian for most cases of interest. | |
| Dec 1 at 4:34 | comment | added | David Gao | As in your last question, since we’re dealing with a separable Hilbert space, direct integral decomposition into factorial representations is always possible, and, unless $\Gamma$ is virtually abelian, you always have some representation that cannot be decomposed into a direct integral of irreducible representations (and that include finite-dimensional factorial representations). This really has nothing to do with whether the group is Fuchsian or Kleinian or anything else. | |
| Dec 1 at 1:45 | history | asked | user82261 | CC BY-SA 4.0 |