An axiomatic version of an elementary operator between two algebras \(\mathcal{A}_1\) and \(\mathcal{A}_2\) (called in this review “an abstract elementary operator”) was introduced by M. Brešar and P. Šemrl [Proc. R. Soc. Edinb., Sect. A, Math. 129, No. 6, 1115–1135 (1999; Zbl 0972.47022)]: an abstract elementary operator of length one is a pair \((M,M^*)\) of linear mappings \(M : \mathcal{A}_1\rightarrow\mathcal{A}_2\) and \(M^* : \mathcal{A}_2\rightarrow\mathcal{A}_1\) satisfying the conditions \(M(xM^*(y)z) = M(x)yM(z)\) and \(M^*(yM(x)u) = M^*(y)xM^*(u)\) for all \(x,z\in\mathcal{A}_1\) and \(y,u\in \mathcal{A}_2\). Abstract elementary operators generalise double centralisers, algebraic isomorphisms and, in the case \(\mathcal{A}_1 = \mathcal{A}_2\), elementary operators on \(\mathcal{A}_1\). In the same paper, it was shown that any abstract elementary operator of length one \((M,M^*)\) between two standard operator algebras is of the form \((M_{S,T}, M_{T,S})\) where \(M_{S,T}(X) = SXT\) (and \(M_{T,S}(Y) = TYS\)) and \(T\) and \(S\) are bounded linear operators between the Banach spaces on which the given algebras act.
The main result in the paper under review is an analogous description of the abstract elementary operators of length one between \(\mathcal{J}\)-subspace lattice algebras. These algebras include the reflexive algebras of pentagon lattices and of atomic Boolean lattices. It is shown that if \((M,M^*)\) is such an operator and if \(M\) and \(M^*\) are surjective, then there exist (in general unbounded) closed densely defined injective operators \(S\) and \(T\) between the underlying Banach spaces such that \(M = M_{S,T}\) and \(M^* = M_{T,S}\) on the domains of \(T\) and \(S\), respectively.