Let $F/\mathbb Q$ be a finite normal extension of the rational numbers. Let $V$ be an $F$-vector space of countably infinite dimension, and set $L=GL\_F(V)$. Put moreover $L^\*$ be the set of all mappings fixing elementwise at least one subspace of finite codimension in $V$.


1) Is $U=L' L^\*/L^\*$ simple modulo its center? What is the center? I heard in a conference that this is the multiplicative group of $F$, but is there any way to easily see it? My guess is that $U$ is an union of special linear groups, but also then why the center is such? 


2) How does $H = \mathrm{Gal}(F/\mathbb Q)$ act on $Z(U)$?



On the general linear group of a vector space of infinite dimension
Everything can be answered relying on the paper *Rosenberg, Alex
The structure of the infinite general linear group. 
Ann. of Math. (2) 68 1958 278-294* ([MR link](https://mathscinet.ams.org/mathscinet-getitem?mr=102528), [JSTOR](https://www.jstor.org/stable/1970248); sorry these are under paywalls.)


Namely, $L$ is perfect, and every proper normal subgroup of $L$ is contained in $L^\*F^\times$. In particular, $U=L/L^+F^\times$ is a simple group. In particular, $F^\times L^\*/L^\*\simeq F^\times$ is equal to the center of $L^\*$.


The action of the Galois group on $F^\times$ is the natural one, and hence this is the same when viewing $F^\times$ as center of $L/L^\*$.


