Graph C ^∗ -algebras, and their relationship to Leavitt path algebras

© Springer-Verlag London Ltd. 2017. In this chapter we investigate the connections between Leavitt path algebras (with coefficients in ℂ), and their analytic counterparts, the graph C ∗ -algebras. We start by giving a brief overview of graph C ∗ -algebras, and then show how the Leavitt path algebra L ℂ (E) naturally embeds as a dense ∗-subalgebra of the graph C ∗ -algebra C ∗ (E). We analyze the structure of the closed ideals in C ∗ (E) for row-finite graphs, and compare this structure to the ideal structure of the corresponding Leavitt path algebra L K (E). We finish the chapter by considering numerous properties which are simultaneously shared by C ∗ (E) and L ℂ (E).