MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

10 deleted 453 characters in body

Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and $$\operatorname{diam} H/\Gamma\le 1.$$ Is it true that $H/\Gamma$ is compact?

Comment.

At the moment I do not have an answer even if $\Gamma$ acts by translations. Here is a related question:

Let $L$ be a lattice in $\mathbb R^q$ ($q$ is any positive integer). Assume $$\operatorname{diam} \mathbb R^q/L>1000.$$ Is it true that there is a midpoint $m$ of two points in $L$ such that $|m-x|>1$ for any $x\in L$?

If the answer to the this question is "YES" then the answer to my original question is "NO".

Stupid example. Assume the action of $\Gamma$ on $H=\ell_2$ is generated by coordinate translations $x_n\mapsto x_n+\epsilon_n$. Then $$\operatorname{diam} H/\Gamma=\tfrac12\cdot\sqrt{\sum_{n=1}^\infty\epsilon_n^2}.$$ Thus, if $\operatorname{diam} H/\Gamma\le 1$ then $H/\Gamma$ is a quotient of Hilbert cube, and has to be compact.

9 added 10 characters in body

Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and $$\operatorname{diam} H/\Gamma\le 1.$$ Is it true that $H/\Gamma$ is compact?

Comment.

At the moment I do not have an answer even if $\Gamma$ acts by translations. Here is a related question:

Let $L$ be a lattice in $\mathbb R^q$ ($q$ is any positive integer). Assume $$\operatorname{diam} H/L>1000.$$ \mathbb R^q/L>1000.$$Is it true that there is a midpoint m of two points in L such that |m-x|>1 for any x\in L? If the answer to the this question is "YES" then the answer to my original question is "NO". Stupid example. Assume the action of \Gamma on H=\ell_2 is generated by coordinate translations x_n\mapsto x_n+\epsilon_n. Then$$\operatorname{diam} H/\Gamma=\tfrac12\cdot\sqrt{\sum_{n=1}^\infty\epsilon_n^2}.$$Thus, if \operatorname{diam} H/\Gamma\le 1 then H/\Gamma is a quotient of Hilbert cube, and has to be compact. 8 ? Assume \Gamma acts by isometries on a separable Hilbert space H, and$$\operatorname{diam} H/\Gamma\le 1.$$Is it true that H/\Gamma is compact? Comment. At the moment I do not have an answer even if \Gamma acts by translations. Here is a related question: Let L be a lattice in \mathbb R^q (q is any positive integer). Assume$$\operatorname{diam} H/L>1000.$$Is it true that there is a midpoint m of two points in L such that |m-x|>1 for any x\in L.? If the answer to the this question is "YES" then the answer to my original question is "NO". Stupid example. Assume the action of \Gamma on H=\ell_2 is generated by coordinate translations x_n\mapsto x_n+\epsilon_n. Then$$\operatorname{diam} H/\Gamma=\tfrac12\cdot\sqrt{\sum_{n=1}^\infty\epsilon_n^2}. Thus, if $\operatorname{diam} H/\Gamma\le 1$ then $H/\Gamma$ is a quotient of Hilbert cube, and has to be compact.

7 added 33 characters in body
6 edited tags
5 added 401 characters in body
4 added 55 characters in body
3 added 306 characters in body
2
1