# Are convolution algebras ever "topologically noetherian"?

For finite groups $G$, we have the group ring $k[G]$, and we can think of $G$-representations as $k[G]$-modules. It is known that for $G$ virtually polycyclic, $k[G]$ is a Noetherian ring, which means that if $V$ is a finitely generated $G$-rep, then all its submodules are also finitely generated.

If $G$ is a topological group with an interesting topology, then $k[G]$ is almost never what you actually want to look at, since it ignores the topology on $G$. Indeed, what we care about are continuous representations of $G$, and there are various continuous group algebras in this case - $C(G)$, $L^{1,2,∞}(G)$, the algebra of Radon measure $M(G)$, and many variants - with multiplication again given by convolution.

Here the analogous property should be "topologically Noetherian". We want a topologically Noetherian module to be one all of whose closed submodules are topologically finitely generated, and a topologically Noetherian ring to be one all of whose t.f.g. modules are top. Noeth. I'm pretty sure this reduces to the usual property of ACC on closed ideals, but if there's any subtlety, this seems like the right definition.

So, are any of the various group algebras (absolutely including any not mentioned above) topologically Noetherian in this sense for nice $G$, say for $G$ a compact Lie group? My sense is that this is probably too much to hope for.

• When you say "closed", are you assuming your convolution algebra is complete in an appropriate topology? Oct 29 '14 at 23:05
• Good point - hm, I guess so? I mean, really what I care about is that the category of continuous G-reps is Noetherian in the sense described. But since things are much more commonly phrased in terms of the ring, I figured people might know more about it from this angle. Oct 31 '14 at 4:59
• In a separable C*-algebra, each closed left/right ideal is topologically singly generated. Nov 25 '14 at 13:53

The completed group ring of a compact $$p$$-adic Lie group with coefficients in a field of characteristic $$p$$ is topologically Noetherian. In fact, it is even abstractly Noetherian. This is a theorem of M. Lazard from the 1960s, and you can read more about these rings here.