Algebraic bivariant K-theory and Leavitt path algebras

Abstract

We investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras L.E/ and L.F / of graphs E and F over a commutative ground ring `. We approach this by studying the structure of such algebras under bivariant algebraic K-theory kk, which is the universal homology theory with the properties above. We show that under very mild assumptions on `, for a graph E with finitely many vertices and reduced incidence matrix AE, the structure of L.E/ in kk depends only on the groups Coker.I AE/ and Coker.I A t E/. We also prove that for Leavitt path algebras, kk has several properties similar to those that Kasparov’s bivariant K-theory has for C -graph algebras, including analogues of the Universal coefficient and Künneth theorems of Rosenberg and Schochet