The center of a Leavitt path algebra

Abstract

In this paper the center of a Leavitt path algebra is computed for a wide range of situations. A basis as a $K$-vector space is found for $Z(L(E))$ when $L(E)$ enjoys some finiteness condition such as being artinian, semisimple, noetherian and locally noetherian. The main result of the paper states that a simple Leavitt path algebra $L(E)$ is central (i.e. the center reduces to the base field $K$) when $L(E)$ is unital and has zero center otherwise. Finally, this result is extended, under some mild conditions, to the case of exchange Leavitt path algebras.