Decomposable Leavitt path algebras for arbitrary graphs

For any field $K$ and for a completely arbitrary graph $E$, we characterize the Leavitt path algebras $L_K(E)$ that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. When the algebra decomposes, it actually does so as a direct sum of Leavitt path algebras for some suitable graphs. Under certain finiteness conditions, a unique indecomposable decomposition exists.
Comment: Forum Math. (27)2015. arXiv admin note: text overlap with arXiv:1207.3466 by other authors