Locally finite Leavitt path algebras

Abstract

A group-graded K-algebra A = ⊕g∈GAg is called locally finite in case each graded component Ag is finite dimensional over K. We characterize the graphs E for which the Leavitt path algebra LK(E) is locally finite in the standard ℤ-grading. For a locally finite ℤ-graded algebra A we show that, if every nonzero graded ideal has finite codimension in A, then every nonzero ideal has finite codimension in A; that is, ℤ-graded just infinite implies just infinite. We use this result to characterize the finite graphs E for which the Leavitt path algebra LK(E) is locally finite just infinite. We then give an explicit description of the graphs and algebras which arise in this way. In particular, we show that the locally finite Leavitt path algebras are precisely the noetherian Leavitt path algebras.