2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/124615We prove Leavitt path algebra versions of the two uniqueness theorems of graph C*-algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras and give necessary and sufficient conditions for their simplicity. We also use these results to give a proof of the fact that for any graph E the Leavitt path algebra $L_\mathbb{C}(E)$ embeds as a dense *-subalgebra of the graph C*-algebra C*(E). This embedding has consequences for graph C*-algebras, and we discuss how we obtain new information concerning the construction of C*(E).34 pages, uses XY-pic. New version comments: Some small typos corrected. This is the final version to appear in the Journal of AlgebraOperator AlgebrasRings and Algebras16W50, 46L55text