Using Steinberg algebras to study decomposability of Leavitt path   algebras
  Given an arbitrary graph $E$ we investigate the relationship between $E$ and the groupoid $G_E$.   We show that there is a lattice isomorphism between the lattice of pairs $(H, S)$, where $H$ is a hereditary and saturated set of vertices and $S$ is a set of breaking vertices {associated to $H $}, onto the lattice of open invariant subsets of $G_E^{(0)}$. We use this lattice isomorphism to characterize the decomposability of the Leavitt path algebra $L_K(E)$, where $K$ is a field.   First we find a graph condition to characterise when an open invariant subset of $G_E^{(0)}$ is closed.   Then we give both a graph condition and a groupoid condition each of which is equivalent to $L_K(E)$ being decomposable {in the sense that it can be written as a direct sum of two nonzero ideals}. 
Lisa Orloff Clark, Dolores Martin Barquero, Candido Martin Gonzalez,   Mercedes Siles Molina
math.RA