Algebraic bivariant K-theory and Leavitt path algebras

摘要

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 l. 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 l, 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 CokerI (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*-graph algebras, including analogues of the Universal coefficient and Kunneth theorems of Rosenberg and Schochet.