The semisimplicity problem for 𝑝-adic group algebras

Abstract

For a prime p p let Ω = Ω p \Omega = {\Omega _p} denote the completion of the algebraic closure of the field of p p -adic numbers with p p -adic valuation | | \left | \right | . Given a group G G consider the ring of formal sums \[ l 1 ( Ω , G ) = { ∑ x ∈ G α x x : α x ∈ Ω , | α x | → 0 } . {l_1}\left ( {\Omega ,G} \right ) = \left \{ {\sum \limits _{x \in G} {{\alpha _x}x:{\alpha _x} \in \Omega ,\left | {{\alpha _x}} \right |} \to 0} \right \}. \] Motivated by the study of group rings and the complex Banach algebras l 1 ( C , G ) {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 ( Ω , G ) G,{l_1}\left ( {\Omega ,G} \right ) is semisimple if and only if G G does not contain a C p ∞ {C_p}\infty subgroup. We also prove that l 1 ( Ω , G ) {l_1}\left ( {\Omega ,G} \right ) is semisimple if G G is a nilpotent p ′ p’ -group, an ordered group, or a torsion-free solvable group. We use a mixture of algebraic and analytic methods.