Corners of Leavitt path algebras of finite graphs are Leavitt path algebras

Abstract: We achieve an extremely useful description (up to isomorphism) of the Leavitt path algebra $L_K(E)$ of a finite graph $E$ with coefficients in a field $K$ as a direct sum of matrix rings over $K$, direct sum with a corner of the Leavitt path algebra $L_K(F)$ of a graph $F$ for which every regular vertex is the base of a loop. Moreover, in this case one may transform the graph $E$ into the graph $F$ via some step-by-step procedure, using the "source elimination" and "collapsing" processes. We use this to establish the main result of the article, that every nonzero corner of a Leavitt path algebra of a finite graph is isomorphic to a Leavitt path algebra. Indeed, we prove a more general result, to wit, that the endomorphism ring of any nonzero finitely generated projective $L_K(E)$-module is isomorphic to the Leavitt path algebra of a graph explicitly constructed from $E$. Consequently, this yields in particular that every unital $K$-algebra which is Morita equivalent to a Leavitt path algebra is indeed isomorphic to a Leavitt path algebra.