Let $\mathcal{H}$ be a separable Hilbert space and $L_{0}\subset B(\mathcal{H} )$ a complete reflexive lattice. Let $\mathcal{K}$ be the direct sum of $n_0$ copies of $\mathcal{H}$ ($n_{0}\in\mathbb{N}$ and $n_0\geq 3$). We construct a class of subspace lattices $L$ on $\mathcal{K}$. Let $Alg(L)$ be the corresponding subspace lattice algebras. We first show that $Alg(L)$ is decomposable if and only if $Alg(L_{0})$ is decomposable. Then we show that an operator $T$ in $Alg(L)$ is single if and only if $T$ is of rank 1 under certain conditions. Finally, we show that every linear local derivation on $Alg(L)$ is a derivation.
When $\mathcal{L}_{\xi}$ be a subspace lattice of one-point extension of a nest on $\mathcal{H}$ and $Alg(\mathcal{L}_{\xi})$ the corresponding Kadison-Singer algebra, we show that every local derivation from $Alg(\mathcal{L}_{\xi})$ into $B(\mathcal{H})$ is a derivation and every derivation from $Alg(\mathcal{L}_{\xi})$ into itself is continuous.
This is based on joint work with Hongjie Chen and Zhujun Yan.
王利广，曲阜师范大学数学科学学院教授，博士生导师。2005年7月于中国科学院获理学博士学位。研究方向为算子代数。目前主持一项国家自然科学基金面上项目，主持完成国家自然科学基金面上项目2项、数学天元基金1项和山东省自然科学基金面上项目2项，参与国家自然科学基金面上项目2项。已在《J.Functional Analysis》、《J. Operator Theory》等期刊发表论文30余篇。
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