Filtered K-theory for graph algebras

Abstract

We introduce filtered algebraic $K$-theory of a ring $R$ relative to a sublattice of ideals. This is done in such a way that filtered algebraic $K$-theory of a Leavitt path algebra relative to the graded ideals is parallel to the gauge invariant filtered $K$-theory for graph $C^*$-algebras. We apply this to verify the Abrams-Tomforde conjecture for a large class of finite graphs.