Non-archimedean group algebras

The group ring R(G) of a group G over a coefficient ring R is well known and so is the L¹ group algebra

$\text{l}{}_1\text{(G)=}\text{∑}\text{g}\text{α}{}_{\mathrm{g}}\text{g:α∈C,∑|α}{}_{\mathrm{g>}}\text{|<∞}$

. We study in this note

$\text{l(R,G)=}\text{∑}\text{g}\text{α}{}_{\mathrm{g}}\text{g:α}{}_{\mathrm{g}}\text{∈R,g∈G,limα}{}_{\mathrm{g}}\text{=0}$

. where R is Z_p(Q_p) the ring (field) of p-adic integers (numbers) equipped with the p-adic valuation. Analogues of certain well known results for group rings and l₁(G) are obtained for l(R,G).