一个乘积不能延拓到其Dedekind完备化的$d$-代数
A $d$-algebra whose product does not extend to its Dedekind completion
- Research Laboratory of Algebra, Topology, Arithmetic, and Order, Department of Mathematics, Faculty of Mathematical, Physical and Natural Sciences of Tunis, Tunis El Manar University, Tunisia（突尼斯艾曼努尔大学数学、物理与自然科学学院数学系代数、拓扑、算术与序研究实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
构造具有强序单位的Banach格$d$-代数，其乘积无法延拓到Dedekind完备化，否定Wickstead和Huijsmans的相关问题及Chil的延拓定理。
AI中文摘要：
我们构造了一个具有强序单位的Banach格$d$-代数，其乘积没有正的结合双线性延拓到其Dedekind完备化。这给出了Wickstead问题的一个否定答案，也给出了Huijsmans问题在$d$-代数情形下的否定答案。它还否定了Chil的一个延拓定理。该例子表明，范数完备性和强序单位并不能确保乘法延拓到Dedekind完备化。
英文摘要：
We construct a Banach lattice $d$-algebra with a strong order unit whose product has no positive associative bilinear extension to its Dedekind completion. This gives a negative answer to a question of Wickstead and to the $d$-algebra case of a question of Huijsmans. It also disproves an extension theorem of Chil. The example shows that norm completeness and a strong order unit do not ensure that multiplication extends to the Dedekind completion.
