  By studying certain kind of centralizer algebras of the affine Schur algebra
$\widetilde{S}(n,r)$ we show that $\widetilde{S}(n,r)$ is Noetherian and we
determine its center. Assuming $n\geq r$, we show that $\widetilde{S}(n+1,r)$
is Morita equivalent to $\widetilde{S}(n,r)$, and the Schur functor is an
equivalence under certain conditions.
