Prime spectrum and primitive Leavitt path algebras

In this paper a bijection between the set of prime ideals of a Leavitt path algebra LK(E) and a certain set which involves maximal tails in E and the prime spectrum of K[x, x −1] is established. Necessary and sufficient conditions on the graph E so that the Leavitt path algebra LK(E) is primitive are also found. introduction Leavitt path algebras of row-finite graphs have been recently introduced in [1] and [7]. They have become a subject of significant interest, both for algebraists and for analysts working in C*-algebras. The Cuntz-Krieger algebras C ∗ (E) (the C*-algebra counterpart of these Leavitt path algebras) are described in [21]. The algebraic and analytic theories, while sharing some striking similarities, they present some remarkable differences, as was shown for instance in the “Workshop on Graph Algebras ” held at the University of Málaga (see [11]), and more deeply in the subsequent enlightening work of Tomforde [23]. For a field K, the algebras LK(E) are natural generalizations of the algebras investigated by Leavitt in [19], and are a specific type of path K-algebras associated to a graph E (modulo certain relations). The family of algebras which can be realized as the Leavitt path algebras