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 $\ell$. 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 $\ell$, for a graph $E$ with finitely many vertices and reduced incidence matrix $A_E$, the structure of $L(E)$ in $kk$ depends only on the groups Coker$(I-A_E)$ and Coker$(I-A_E^t)$. We also prove that for Leavitt path algebras, $kk$ has several properties similar to those that Kasparov's bivariant $K$-theory has for $C^{\ast }$-graph algebras, including analogues of the Universal coefficient and Künneth theorems of Rosenberg and Schochet.