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.