  Let $G$ be a locally profinite group and let $k$ be a field of positive
characteristic $p$. Let $Z(G)$ denote the center of $G$ and let
$\mathfrak{Z}(G)$ denote the Bernstein center of $G$, that is, the $k$-algebra
of natural endomorphisms of the identity functor on the category of smooth
$k$-linear representations of $G$. We show that if $G$ contains an open pro-$p$
subgroup but no proper open centralisers, then there is a natural isomorphism
of $k$-algebras $\mathfrak{Z}(Z(G)) \xrightarrow{\cong} \mathfrak{Z}(G)$. We
also describe $\mathfrak{Z}(Z(G))$ explicitly as a particular completion of the
abstract group ring $k[Z(G)]$. Both conditions on $G$ are satisfied whenever
$G$ is the group of points of any connected smooth algebraic group defined over
a local field of residue characteristic $p$. In particular, when the algebraic
group is semisimple, we show that $\mathfrak{Z}(G) = k[Z(G)]$.
