The semisimplicity problem for $p$-adic group algebras

Abstract:

For a prime $p$ let $\Omega = {\Omega _p}$ denote the completion of the algebraic closure of the field of $p$-adic numbers with $p$-adic valuation $\left | \right |$. Given a group $G$ consider the ring of formal sums \[ {l_1}\left ( {\Omega ,G} \right ) = \left \{ {\sum \limits _{x \in G} {{\alpha _x}x:{\alpha _x} \in \Omega ,\left | {{\alpha _x}} \right |} \to 0} \right \}.\] Motivated by the study of group rings and the complex Banach algebras ${l_1}\left ( {{\mathbf {C}},G} \right )$, we consider the problem of when this ring is semisimple (semiprimitive). Our main result is that for an Abelian group $G,{l_1}\left ( {\Omega ,G} \right )$ is semisimple if and only if $G$ does not contain a ${C_p}\infty$ subgroup. We also prove that ${l_1}\left ( {\Omega ,G} \right )$ is semisimple if $G$ is a nilpotent $p’$-group, an ordered group, or a torsion-free solvable group. We use a mixture of algebraic and analytic methods.