Annihilator ideals of graph algebras

Abstract

If $I$ is a (two-sided) ideal of a ring $R$, we let $\operatorname{ann}_l(I)=\{r\in R\mid rI=0\},$ $\operatorname{ann}_r(I)=\{r\in R\mid Ir=0\},$ and $\operatorname{ann}(I)=\operatorname{ann}_l(I)\cap \operatorname{ann}_r(I)$ be the left, the right and the double annihilators. An ideal $I$ is said to be an annihilator ideal if $I=\operatorname{ann}(J)$ for some ideal $J$ (equivalently, $\operatorname{ann}(\operatorname{ann}(I))=I$). 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 $\operatorname{ann}(I)$ is a graded ideal (with respect to the natural grading of $L_K(E)$ by $\mathbb Z$). We note that $\operatorname{ann}_l(I)$ and $\operatorname{ann}_r(I)$ are also graded. For a graded ideal $I,$ we describe $\operatorname{ann}(I)$ 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.