Local derivation on some class of subspace lattice algebras
Hongjie Chen, Liguang Wang, Zhujun Yang
TL;DR
The paper addresses the question of when local derivations on subspace lattice algebras are genuine derivations. It constructs two reflexive lattices, $\\mathcal{L}_{n_{0}}$ and $\\mathcal{L}_{\\infty}$, from a base lattice $\\mathcal{L}_{0}$ and analyzes their algebras $Alg\\mathcal{L}$, using rank-one and finite-rank operator techniques to extend local derivations to derivations. Under decomposability and the condition $R_{1}(Alg\\mathcal{L}_{0}) \subset Id(Alg\\mathcal{L}_{0})$, the authors prove that every local derivation $\\delta: Alg\\mathcal{L} \to B(\\mathscr{K})$ is a derivation for $\\mathcal{L} \in \\{\\mathcal{L}_{n_{0}}, \\mathcal{L}_{\\infty}\}$. This advances understanding of local derivations in non-self-adjoint operator algebras and connects to Kadison–Singer type algebras.
Abstract
Let $\mathcal{H}$ be a separable Hilbert space and $\mathcal{L}_{0}\subset B(\mathcal{H})$ a complete reflexive lattice. Let $\mathscr{K}$ be the direct sum of $n_0$ copies of $\mathcal{H}$ ($n_{0}\in\mathbb{N}$ and $n_0\geq 2$) or the direct sum of countably infinite many copies of $\mathcal{H}$ respectively. We construct two class of subspace lattices $\mathcal{L}$ on $\mathscr{K}$. Let $Alg\mathcal{L}$ be the corresponding subspace lattice algebra. We show that every local derivation from $Alg\mathcal{L} $ into $B(\mathscr{K})$ is a derivation.