Graded K -theory, filtered K -theory and the classification of graph algebras

We prove that an isomorphism of graded Grothendieck groups K^gr 0 of two Leavitt path algebras induces an isomorphism of a certain quotient of algebraic filtered K-theory and consequently an isomorphism of filtered K-theory of their associated graph C^*-algebras. As an application, we show that since for a finite graph E with no sinks, K^gr₀ (L(E)) of the Leavitt path algebra L(E) coincides with Krieger’s dimension group of its adjacency matrix A_E, our result relates the shift equivalence of graphs to the filtered K-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graph C^*-algebras. This result was only known for irreducible graphs.