Timeline for Kazhdan Property T of semisimple Lie groups
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 22, 2020 at 15:34 | comment | added | A beginner mathmatician | @LSpice. I have understand the identification clearly. But in my question (and also in the paper I mentioned in the question, "$\phi_\lambda(\mu)$" can be defined only for the spherical positive-definite functions $i(\chi)$ (this notation appears in my previous posts.) In this case $\|\mu\|_T=\tau(\mu)$ because the Gelfand transform is an isometry. So how to use your "subset" argument to show $\|\mu\|_T\geq \tau(\mu)$? | |
| Apr 21, 2020 at 19:56 | review | Close votes | |||
| Apr 26, 2020 at 16:44 | |||||
| Apr 21, 2020 at 19:53 | comment | added | A beginner mathmatician | Let us continue this discussion in chat. | |
| Apr 21, 2020 at 19:04 | comment | added | LSpice | $G$ is a (finite-dimensional) Lie group, so it is $\sigma$-compact. | |
| Apr 21, 2020 at 18:48 | comment | added | A beginner mathmatician | Ok. I agree. But here $G$ has to be $\sigma$-compact for that. Right? Since we know that probability measures on a compact Hausdroff space is closed convex hull of Dirac masses. Hence you can take a measure, make it compactly supported, normalized, approximate by convex combinations of Dirac masses and the make the compact sets large so that it covers $G$ asymptotically. | |
| Apr 21, 2020 at 18:41 | comment | added | LSpice | The closed span of the various unit $K g K$'s is $M(G, K)$, so injectivity follows. (Also, you can delete comments.) | |
| Apr 21, 2020 at 18:29 | comment | added | A beginner mathmatician | @LSpice. I understand that given a character $\chi$ of $A_\tau$ I can define a character a character on $G$ as $i(\chi)g\mapsto \tau(m_K*\delta_g*m_k).$ But I think what I need to show that the map $\chi\mapsto i(\chi)$ is injective and continuous in some topology to obtain the spectral theorem. Am I right? Even after the identification you mentioned there are some subtlities. I checked the references. In the book of Gangoli something like that is given but for $L^1(G,K)$. | |
| Apr 21, 2020 at 18:17 | comment | added | LSpice | Anyway, (1) notice that the spectrum is not identified with all pdsf's, but only a subset, and (2) I think that the identification is the simple one: given a character $\chi$ of $\overline{\tau(M(G, K))}$, send $g \in G$ to the value of $\chi \circ \tau$ at the unit mass on $K g K$. | |
| Apr 21, 2020 at 18:07 | comment | added | LSpice | Also, $\phi_\lambda(\mu)$ means that $\phi_\lambda$ is viewed as a character of $\overline{\tau(M(G, K))}$, then evaluated at $\tau(\mu)$. | |
| Apr 21, 2020 at 17:58 | comment | added | LSpice | Did you check the four references given for the identification? | |
| Apr 21, 2020 at 17:58 | history | edited | LSpice | CC BY-SA 4.0 |
Link to paper
|
| Apr 21, 2020 at 17:01 | history | edited | YCor | CC BY-SA 4.0 |
resumed edits erased by collision of edits...
|
| Apr 21, 2020 at 16:26 | history | edited | A beginner mathmatician | CC BY-SA 4.0 |
added 1 character in body
|
| S Apr 21, 2020 at 15:00 | history | suggested | Carl-Fredrik Nyberg Brodda | CC BY-SA 4.0 |
Made reference to paper proper.
|
| Apr 21, 2020 at 12:58 | review | Suggested edits | |||
| S Apr 21, 2020 at 15:00 | |||||
| Apr 21, 2020 at 12:57 | history | edited | YCor | CC BY-SA 4.0 |
edited tags, formatting
|
| Apr 21, 2020 at 12:48 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
added the (property-t) tag
|
| Apr 21, 2020 at 12:44 | history | asked | A beginner mathmatician | CC BY-SA 4.0 |