Local derivation on some class of subspace lattice algebras

Abstract

Let $\mathcal{H}$ be a separable Hilbert space and ${\mathcal{L}}_0\subset B(\mathcal{H})$ a complete reflexive lattice. Let $\mathcal{K}$ be the direct sum of $n_0$ copies of $\mathcal{H}$ ($n_0\in \mathbb{N}$ and $n_0\ge 2$) or the direct sum of countably infinite many copies of $\mathcal{H}$ respectively. We construct two class of subspace lattices $\mathcal{L}$ on $\mathcal{K}$. Let $Alg\mathcal{L}$ be the corresponding subspace lattice algebra. We show that every local derivation from $Alg\mathcal{L}$ into $B(\mathcal{K})$ is a derivation.