GRAPH CHARACTERISATION OF THE ANNIHILATOR IDEALS OF LEAVITT PATH ALGEBRAS

Abstract

If E is a graph and K is a field, we consider an ideal I of the Leavitt path algebra $L_K(E)$ of E over K. We describe the admissible pair corresponding to the smallest graded ideal which contains I where the grading in question is the natural grading of $L_K(E)$ by ${\mathbb {Z}}$ . Using this description, we show that the right and the left annihilators of I are equal (which may be somewhat surprising given that I may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterisation from the title. Then, we turn to the property that the right (equivalently left) annihilator of any ideal is a direct summand and recall that a unital ring with this property is said to be quasi-Baer. We exhibit a condition on E which is equivalent to unital $L_K(E)$ having this property.