A note on the isomorphism conjectures for Leavitt path algebras

摘要

We relate two conjectures which have been raised for classification of Leavitt path algebras. For purely infinite simple unital Leavitt path algebras, it is conjectured that K_0 classifies them completely (Abrams et al., 2008, 2011 [3,4]). For arbitrary unital Leavitt path algebras, it is conjectured that K0gr classifies them completely (Hazrat, in press [12]). We show that for two finite graphs with no sinks (which their associated Leavitt path algebras include the purely infinite simple ones) if their K0gr-groups of their Leavitt path algebras are isomorphic then their K_0-groups are isomorphic as well. We also provide a short proof of the fact that for a finite graph, its associated Leavitt path algebra is strongly graded if and only if the graph has no sinks.