Quasi-Baer $ * $ -Ring Characterization of Leavitt Path Algebras

We say that a graded ring (\( * \)-ring) \( R \) is a graded quasi-Baer ring (graded quasi-Baer \( * \)-ring) if, for each graded ideal \( I \) of \( R \), the right annihilator of \( I \) is generated by a homogeneous idempotent (projection). We prove that a Leavitt path algebra is quasi-Baer (quasi-Baer \( * \)) if and only if it is graded quasi-Baer (graded quasi-Baer \( * \)). We show that a Leavitt path algebra is quasi-Baer (quasi-Baer \( * \)) if its zero component is quasi-Baer (quasi-Baer \( * \)). However, we give some example that showing that the converse implication fails. Finally, we characterize the Leavitt path algebras that are quasi-Baer \( * \)-rings in terms of the properties of the underlying graph.