Centers of Path Algebras, Cohn and Leavitt Path Algebras

This paper is devoted to the study of the center of several types of path algebras associated to a graph E over a field K . First we consider the path algebra KE and prove that if the number of vertices is infinite then the center is zero; otherwise, it is K , except when the graph E is a cycle in which case the center is K [ x ], the polynomial algebra in one indeterminate. Then we compute the centers of prime Cohn and Leavitt path algebras. A lower and an upper bound for the center of a Leavitt path algebra are given by introducing the graded Baer radical for graded algebras. In the final section we describe the center of a prime graph C ∗ -algebra for a row-finite graph.