Isomorphisms and generalized derivations of some algebras (Q607073)

The paper consists of two parts. In the first part, the following theorem is proved. Let \({\mathcal L}\) be a commutative subspace lattice and \({\mathcal A}=\text{alg}\,{\mathcal L}\). If \({\mathcal B}\) is a unital Banach algebra and \(h:~{\mathcal A} \to {\mathcal B}\) is a bounded bijective linear map such that \(h(I)=I\) and \(h(A)h(B)h(C)=0\) for all \(A, B, C\in {\mathcal A}\) with \(AB=BC=0\), then \(h\) is an isomorphism. In the second part, it is proved that for a unital subalgebra \({\mathcal A}\) which is generated by finite-rank operators in alg\(\,{\mathcal L}\), where \({\mathcal L}\) is an \({\mathcal I}\)-subspace lattice, every generalized Jordan derivation from \({\mathcal A}\) to a unital \({\mathcal A}\)-bimodule is actually a generalized derivation.