The centre of completed group algebras of pro-\(p\) groups.
Let \(G\) be a pro-\(p\)-group. The completed group algebra \(\Omega_G=\mathbb{F}_p[\![G]\!]\) of \(G\) over the field \(\mathbb{F}_p\) is defined as the inverse limit \(\varprojlim_{N\vartriangleleft_0G}\mathbb{F}_p[G/N]\). When \(G\) is analytic the \(p\)-adic analogue \(\Lambda_G=\mathbb{Z}_p[\![G]\!]=\varprojlim_{N\vartriangleleft_0G}\mathbb{Z}_p[G/N]\) can also be defined. The author computes the centre of these group algebras when \(G\) is a countably based pro-\(p\) group; he proves that the centre of \(\Omega_G\) (resp. \(\Lambda_G\)) is equal to the closure of the centre of \(\mathbb{F}_p[G]\) (resp. \(\mathbb{Z}_p[G]\)). A cleaner result holds when \(G\) is \(p\)-valued: under this hypothesis \(Z(\Omega_G)=\Omega_Z\) and \(Z(\Lambda_G)=\Lambda_Z\), where \(Z\) is the centre of \(G\).
scientific article; zbMATH DE number 2146814