Ideal-related $K$-theory for Leavitt path algebras and graph $C^*$-algebras

Abstract

We introduce a notion of ideal-related K-theory for rings, and use it to prove that if two complex Leavitt path algebras are equivalent (respectively, isomorphic), then the ideal-related K-theories (respectively, the unital ideal-related K-theories) of the corresponding graph C*-algebras are isomorphic. This has consequences for the Morita equivalence conjecture and isomorphism conjecture for graph algebras, and allows us to prove that when E and F belong to specific collections of graphs whose C*-algebras are classified by ideal-related K-theory, equivalence (respectively, isomorphism) of the Leavitt path algebras implies strong equivalence (respectively, isomorphism) of the graph C*-algebras. We state a number of corollaries that describe various classes of graphs where these implications hold. In addition, we conclude with a classification of Leavitt path algebras of amplified graphs similar to the existing classification for graph C*-algebras of amplified graphs.