发表机构
Iowa State University（爱荷华州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文将Christensen-Pedersen论证推广到任意基数，定义$\kappa$-齐次性，证明$\kappa$-齐次$\mathrm{AW}^*$-代数是$\kappa$-单调完备的，并应用于$\mathrm{AW}^*$-因子得到单调完备性。
AI 中文摘要
Christensen和Pedersen证明了每个真无穷的$\mathrm{AW}^*$-代数是单调序列完备的。我们分离出他们的论证中可推广到任意基数的部分。他们的证明中使用的真无穷$\mathrm{AW}^*$-代数的性质是存在一个正交投影序列，每个投影等价于$1$，且它们的和为$1$。我们通过定义$\kappa$-齐次性的概念来推广这一点，其中$\kappa = \aleph_{0}$的情形恢复了上述性质。$\kappa$-齐次性在超限投影膨胀的每个后继阶段提供所需的新正交空间，而$\mathrm{AW}^*$-代数的正规性在每个极限阶段以及构造结束时提供从投影的并到自伴序中的上确界的过渡。这第二点取代了可数情形中使用的可加性定理和扰动论证。我们证明每个$\kappa$-齐次的$\mathrm{AW}^*$-代数是$\kappa$-单调完备的，并且$\kappa$-单调完备性连同$\kappa$个将正元素与零分离的普通态蕴含单调完备性。作为应用，一个具有忠实态的角的$\mathrm{AW}^*$-因子是单调完备的。
英文摘要
Christensen and Pedersen proved that every properly infinite $\mathrm{AW}^*$-algebra is monotone sequentially complete, and Saitô and Wright developed a transfinite form of their dilation argument. We revisit the transfinite construction using normality of $\mathrm{AW}^*$-algebras. Normality simplifies the limit stages by turning suprema into compressions of joins, so the construction only needs a supply of fresh orthogonal projections large enough to contain the supports of the summands at successor stages. We use this simplified proof to show that a $*$-homomorphism between $\mathrm{AW}^*$-algebras that preserves only the joins needed to encode such a sum preserves the sum itself. We also use it to deduce order-continuity facts about $κ$-join-preserving $*$-homomorphisms. We also show that a finite $\mathrm{AW}^*$-algebra has suprema for all bounded positive families whose supports have bounded total center-valued dimension.
Comments18 pages, rewritten to acknowledge precedents and emphasize novel contributions
