Characterizations of isomorphisms and derivations of some algebras (Q884379)

Let \(\mathcal{L}\) be a commutative subspace lattice on a Hilbert space \(H\). This is a collection of subspaces of \(H\) with \(\{0\}\), \(H\) in \(\mathcal{L}\), both the intersection and the closed linear span of every family of elements in \(\mathcal{L}\) belong to \(\mathcal{L}\), and the projections of \(H\) onto the subspaces of \(\mathcal{M}\) commute with each other. The authors show that if \(\phi\) is a continuous bijective linear map from the Banach algebra \(\text{alg}(\mathcal{L})=\{ T\in B(H)\mid TN\subset N,\;N\in\mathcal{L}\}\) onto any unital Banach algebra \(B\) with the property that \(\phi(a)\phi(b)=0\) whenever \(a,b\in \text{alg}(\mathcal{L})\) are such that \(ab=0\), then \(\phi(\mathbf{1})\) is an invertible element in the centre of \(B\) and \(\phi(\mathbf{1})^{-1}\phi\) is an isomorphism from \(A\) onto \(B\). The authors also characterize all linear maps \(d\) from a nest algebra \(A\) into a factor von Neumann algebra \(B\) with the property that \(d(a)b+ad(b)=0\) whenever \(a,b\in A\) are such that \(ab=0\).