  We introduce some modified forms for the degenerate and non-degenerate affine
Hecke algebras of type $A$. These are certain subalgebras living inside the
inverse limit of cyclotomic Hecke algebras. We construct faithful
representations and standard bases for these algebras and give some explicit
description of their centers. We show that there are algebra isomorphisms
between some generalized Ore localizations of these modified affine Hecke
algebras and of the quiver Hecke algebras of type $A$. As an application, we
show that the center conjecture for the cyclotomic quiver Hecke algebra of type
$A$ holds if and only if the center conjecture for the cyclotomic Hecke algebra
of type $A$ holds.
