Ring isomorphisms and pentagon subspace lattices. (Q1873698)

Let \(X\) be a Banach space and \(K\), \(L\) and \(M\) be subspaces of \(X\) such that \(L\subset M\), \(K\cap M = \{0\}\) and \(K\vee L = X\). Then the collection \(\mathcal{P} = \{\{0\}, K, L, M, X\}\) is called a pentagon lattice on \(X\). If \(\mathcal{P}\) is a pentagon lattice on \(X\), let \(\text{Alg} \mathcal{P}\) be the algebra of bounded linear operators on \(X\) leaving the elements of \(\mathcal{P}\) invariant. Fix pentagon lattices \(\mathcal{P}_1\) and \(\mathcal{P}_2\) on Banach spaces \(X_1\) and \(X_2\), respectively. It was shown in [\textit{A. Katavolos}, \textit{M. S. Lambrou} and \textit{W. E. Longstaff}, J. Oper. Theory 46, 355--380 (2001; Zbl 0998.47045)] that every algebra isomorphism \(\Phi\) between \(\text{Alg} \mathcal{P}_1\) and \(\text{ Alg} \mathcal{P}_2\) is quasi-spacially induced in the sense that there exists a densely defined closed injective linear operator \(T : \mathcal{D}\rightarrow X_2\) (\(\mathcal{D}\subset X_1\)) with dense range such that \(\Phi(A)Tx = TAx\) for all \(x\in X_1\) and \(A\in \text{Alg} \mathcal{P}_1\). In the paper under review, the authors study ring isomorphisms \(\Phi\) between \(\text{Alg} \mathcal{P}_1\) and \(\text{Alg} \mathcal{P}_2\). Their main result is that every such \(\Phi\) is either a quasi-spatially induced linear or a quasi-spatially induced conjugate-linear algebra isomorphism.