Decompositions of Infinite-Dimensional $A_{\mathrm{\infty },\mathrm{\infty }}$ Quiver Representations

Using linear algebraic methods we show that every (possibly infinite-dimensional) representation of a quiver with underlying graph $A_{\mathrm{\infty },\mathrm{\infty }}$ is Krull-Schmidt, as long as the arrows in the quiver eventually point outward.