ar X iv : 1 50 7 . 07 43 9 v 2 [ m at h . R A ] 4 N ov 2 01 5 CENTERS OF LEAVITT PATH ALGEBRAS AND THEIR COMPLETIONS

In [8, 9] M. G. Corrales Garcia, D. M. Barquero, C. Martin Gonzalez, M. Siles Molina, J. F Solanilla Hernandez described the center of a Leavitt path algebra and characterized it in terms of the underlying graph. We offer a different characterization of the center. In particular, we prove that the Boolean algebra of central idempotents of a Leavitt path algebra of a finite graph is isomorphic to the Boolean algebra of finitary annihilator hereditary subsets of the graph. 1. Definitions and The Main Results A (directed) graph Γ = (V,E, s, r) consists of two sets V and E that are respectively called vertices and edges, and two maps s, r : E → V. The vertices s(e), r(e) are referred to as the source and the range of the edge e respectively. The graph is called row-finite if for all vertices v ∈ V , card(s(v)) < ∞. A vertex v for which s(v) = φ is called a sink. A path p = e1 · · · en of length l(p) = n in a graph Γ is a sequence of edges e1,. . . ,en such that r(ei) = s(ei+1) for i = 1, 2, . . . , n − 1. In this case we say that s(p) = s(e1), r(p) = r(en). If W ⊂ V and r(p) ∈ W then the path p is said to be a path from the vertex s(p) to the subset W . Let Path(Γ) denote the set of all paths in the graph Γ. If s(p) = r(p) then the path p is closed. If p = e1 · · · en is closed path and the vertices s(e1), . . . , s(en) are distinct then we call the path p a cycle. Denote V (p) = {s(e1), . . . , s(en)}, E(p) = {e1,. . . ,en}. An edge e ∈ E is called an exit of a cycle C if s(e) ∈ V (C), but e / ∈ E(C). A cycle that has no exits is called a NE-cycle. We say that a vertex w is a descendant of a vertex v if there exists a path p ∈ Path(Γ) such that s(p) = v, r(p) = w.