Elementary operators on strongly double triangle subspace lattice algebras (Q710884)

The concept of abstract elementary operators was introduced by \textit{M.\,Brešar} and \textit{P.\,Šemrl} in [Proc.\ R.\ Soc.\ Edinb., Sect.\ A, Math.\ 129, 1115--1135 (1999; Zbl 0972.47022)]. Let \(\mathcal A\) and \(\mathcal B\) be complex algebras. An abstract elementary operator of length one between \(\mathcal A\) and \(\mathcal B\) is a pair of linear operators \(M: \mathcal A \to \mathcal B\) and \(M^*: \mathcal B \to \mathcal A\) satisfying \[ M(x(M^*y)z)= (Mx)y(Mz) \quad \text{and}\quad M^* (y (Mx) t)= (M^*y)x (M^*t) \] for all \(x,z \in \mathcal A\), \(y, t \in \mathcal B\). On the other hand, let \(X\) be a nonzero reflexive complex Banach space. A strongly double triangle subspace lattice on \(X\) is a set \(\mathcal D=\{ \{ 0 \}, K, L, M, X \} \) of subspaces of \(X\) satisfying \(K \cap L= L \cap M= M \cap K = \{ 0 \} \) and \(\overline { K+ L } = \overline{L+ M}= \overline{M+K}=X\) (where \(\overline{K+L}\) denotes the closure of \(K+L\)). As usual, the associated algebra is \(\text{Alg}\,\mathcal D\), the set of bounded linear operators on \(X\) that leave every member of \(\mathcal D\) invariant. In the paper under review, the authors study surjective elementary operators between two strongly double triangle subspace lattice algebras on reflexive Banach spaces \(X_1\) and \(X_2\). It should be pointed out that algebraic isomorphisms between strongly double triangle subspace lattice algebras were studied in [\textit{Y.-F.\thinspace Pang} and \textit{G.-X.\thinspace Ji}, Linear Algebra Appl.\ 422, No.\,1, 265--273 (2007; Zbl 1116.47057)].