Algebras of Quotients of Leavitt Path Algebras

We start this paper by showing that the Leavitt path algebra of a (row-finite) graph is an algebra of quotients of the corresponding path algebra. The path algebra is semiprime if and only if whenever there is a path connecting two vertices, there is another one in the opposite direction. Semiprimeness is studied because, for acyclic graphs, the Leavitt path algebra is a Fountain-Gould algebra of right quotients of any semiprime subalgebra containing the path algebra (and a Moore-Penrose algebra of right quotients of any subalgebra with involution that contains the path algebra). The maximal algebras of quotients of Leavitt path algebras with essential socle (equivalently the associated graph satisfies that every vertex connects to a line point) can be obtained. We also described the algebraic counterpart of the Toeplitz algebras.