Using the Steinberg algebra model to determine the center of any Leavitt path algebra

TL;DR

This paper characterizes the center of Leavitt path algebras over any graph using Steinberg algebra models, linking algebraic properties to graph-theoretic conditions, and provides a basis for the center.

Contribution

It introduces a novel approach using Steinberg algebra models to determine the center of Leavitt path algebras for arbitrary graphs.

Findings

Characterization of compact open invariant subsets via Condition (F)

Basis of the center depending on minimal compact open invariant subsets

Connection between graph properties and algebraic center structure

Abstract

Given an arbitrary graph, we describe the center of its Leavitt path algebra over a commutative unital ring. Our proof uses the Steinberg algebra model of the Leavitt path algebra. A key ingredient is a characterization of compact open invariant subsets of the unit space of the graph groupoid in terms of the underlying graph: an open invariant subset is compact if and only if its associated hereditary and saturated set of vertices satisfies Condition (F). We also give a basis of the center. Its cardinality depends on the number of minimal compact open invariant subsets of the unit space.