论文标题
局部稳定的收敛和正算子的订单连续性
Locally solid convergences and order continuity of positive operators
论文作者
论文摘要
我们认为具有局部固体收敛结构的矢量晶格，这些结构不一定是拓扑。我们表明，这种融合是由阳性锥体上的融合到$ 0 $的定义。仅在部分情况下可用的无界修饰结果是普遍的。阶收敛的特征是最强的局部固体收敛性，其中单调网将收敛到其极值（如果存在）。我们部分地表征了sublattices，在这种sublatices上，订单收敛是对环境晶格的订单收敛的限制。我们表明，同态是连续的，如果是连续的，则是连续的。 UO收敛的特征是独立于阶收敛的特征。我们表明，在连续函数的空间上，UO收敛弱于紧凑的开放收敛，如果基础拓扑空间包含一个密集的局部紧凑子空间。对于一大批融合，我们证明正常运算符在且仅当其限制到密集的常规sublattice是连续的秩序时是连续的，而常规sublattice的闭合是常规的，而原始的sublattice在关闭中是密集的。我们还提供了一个定期的sublattice的示例，其局部拓扑矢量晶格的闭合不规律。
We consider vector lattices endowed with locally solid convergence structures, which are not necessarily topological. We show that such a convergence is defined by the convergence to $0$ on the positive cone. Some results on unbounded modification which were only available in partial cases are generalized. Order convergence is characterized as the strongest locally solid convergence in which monotone nets converge to their extremums (if they exist). We partially characterize sublattices on which the order convergence is the restriction of the order convergence of the ambient lattice. We show that homomorphism is order continuous iff it is uo-continuous. Uo convergence is characterized independently of order convergence. We show that on the space of continuous function uo convergence is weaker than the compact open convergence iff the underlying topological space contains a dense locally compact subspace. For a large class of convergences we prove that a positive operator is order continuous if and only if its restriction to a dense regular sublattice is order continuous, and that the closure of a regular sublattice is regular with the original sublattice being order dense in the closure. We also present an example of a regular sublattice of a locally solid topological vector lattice whose closure is not regular.
