Annihilator ideals of graph algebras

Abstract

If I is a (two-sided) ideal of a ring R, we let $${\text {ann}}_l(I)=\{r\in R\mid rI=0\},$$ $${\text {ann}}_r(I)=\{r\in R\mid Ir=0\},$$ and $${\text {ann}}(I)={\text {ann}}_l(I)\cap {\text {ann}}_r(I)$$ be the left, the right and the double annihilators. An ideal I is said to be an annihilator ideal if $$I={\text {ann}}(J)$$ for some ideal J (equivalently, $${\text {ann}}({\text {ann}}(I))=I$$ ). We study annihilator ideals of Leavitt path algebras and graph $$C^*$$ -algebras. Let $$L_K(E)$$ be the Leavitt path algebra of a graph E over a field K. If I is an ideal of $$L_K(E),$$ it has recently been shown that $${\text {ann}}(I)$$ is a graded ideal (with respect to the natural grading of $$L_K(E)$$ by $$\mathbb Z$$ ). We note that $${\text {ann}}_l(I)$$ and $${\text {ann}}_r(I)$$ are also graded. If I is graded, we show that $${\text {ann}}_l(I)={\text {ann}}_r(I)={\text {ann}}(I)$$ and describe $${\text {ann}}(I)$$ in terms of the properties of a pair of sets of vertices of E, known as an admissible pair, which naturally corresponds to I. Using such a description, we present properties of E which are equivalent with the requirement that each graded ideal of $$L_K(E)$$ is an annihilator ideal. We show that the same properties of E are also equivalent with each of the following conditions: (1) the lattice of graded ideals of $$L_K(E)$$ is a Boolean algebra; (2) each closed gauge-invariant ideal of $$C^*(E)$$ is an annihilator ideal; (3) the lattice of closed gauge-invariant ideals of $$C^*(E)$$ is a Boolean algebra. In addition, we present properties of E which are equivalent with each of the following conditions: (1) each ideal of $$L_K(E)$$ is an annihilator ideal; (2) the lattice of ideals of $$L_K(E)$$ is a Boolean algebra; (3) each closed ideal of $$C^*(E)$$ is an annihilator ideal; (4) the lattice of closed ideals of $$C^*(E)$$ is a Boolean algebra.