Let's consider general linear group $GL(V)$. It's well known that if $\dim V<\infty$ then the center of this group is the subgroup of scalar operators. Is this also a case when $\dim V=\infty$?
Asked
Active
Viewed 44 times
0
-
1I am not sure if this matters, but I would guess it does. What exact cardinality are you talking about? Which infinity? – Malady Feb 09 '24 at 16:29
-
Wouldn't the default finite-dimensional proof (with matrix units) be meaningful no matter the cardinality? – Amateur_Algebraist Feb 09 '24 at 16:41
-
And what is $V$ and $GL(V)$? More precisely, are you working with vector spaces or topological vector spaces, etc.? – Moishe Kohan Feb 09 '24 at 16:41
-
@MoisheKohan well by $GL(V)$ I mean invertible lineat transformations of vector space, without any additional structure – Big Coconut Feb 09 '24 at 16:43
-
@MoisheKohan and I found out that $GL(\infty)$ is actually not the same thing – Big Coconut Feb 09 '24 at 16:43