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The semisimplicity problem for $ p$-adic group algebras


Authors: Kathryn E. Hare and Maziar Shirvani
Journal: Proc. Amer. Math. Soc. 108 (1990), 653-664
MSC: Primary 46S10; Secondary 16D60, 22D15
DOI: https://doi.org/10.1090/S0002-9939-1990-0998736-0
MathSciNet review: 998736
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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\vert \right\vert$. Given a group $ G$ consider the ring of formal sums

$\displaystyle {l_1}\left( {\Omega ,G} \right) = \left\{ {\sum\limits_{x \in G} ... ...:{\alpha _x} \in \Omega ,\left\vert {{\alpha _x}} \right\vert} \to 0} \right\}.$

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.

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DOI: https://doi.org/10.1090/S0002-9939-1990-0998736-0
Article copyright: © Copyright 1990 American Mathematical Society

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