Uniqueness theorems and ideal structure for Leavitt path algebras

We 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 LC(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).