A note on the core of Steinberg algebras

Abstract: For an ample Hausdorff groupoid $G$, and the Steinberg algebra $A_R(G)$ with coefficients in the commutative ring $R$ with unit, we describe the centraliser of subalgebra $A_R(U)$ with $U$ an open closed invariant subset of unit space of $G$. In particular, we obtain that the algebra of the interior of the isotropy is indeed the centraliser of the diagonal subalgebra of Steinberg algebra. This will unify several results in the literature and the corresponding results for Leavitt path algebras follow.