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

TL;DR

This paper develops a new ideal-related K-theory framework for rings and uses it to connect Morita equivalence and isomorphism of Leavitt path algebras with the classification of graph C*-algebras, advancing the understanding of their structural relationships.

Contribution

It introduces ideal-related K-theory for rings and applies it to establish links between algebraic and C*-algebraic classifications of graph algebras.

Findings

Morita equivalence of Leavitt path algebras implies isomorphism of ideal-related K-theories of graph C*-algebras.

Certain classes of graphs have their Leavitt path algebra equivalences reflected in the C*-algebra classification.

Classification results for Leavitt path algebras of amplified graphs are obtained, similar to known C*-algebra classifications.

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 Morita 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, Morita equivalence (respectively, isomorphism) of the Leavitt path algebras implies strong Morita 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.