Isomorphisms and generalized derivations of some algebras

For a commutative subspace lattice L on a complex Hilbert space and a bounded bijective linear mapping h from algL onto a unital Banach algebra B, we show that if h satisfies h(A)h(B)h(C)=0 for all A,B,C in algL with AB=BC=0 and h(I)=I, then h is an isomorphism. For a J‐subspace lattice L on a Banach space and the unital subalgebra A of algL generated by finite-rank operators, we show that all generalized Jordan derivations from A to any unital A‐bimodule are generalized derivations.