几乎 \\(f\\)-代数的张量积：一个反例

中文摘要 AI 辅助

我们证明两个阿基米德几乎 \\(f\\)-代数的自然张量积乘法一般不扩展到它们的 Fremlin 张量积。反例中的两个代数具有相同的底层向量格 \\(C[0,1]\\)。它们的乘法是显式的、正的且指数为三的幂零。障碍在于函数 \\((s,t)\mapsto e^{st}\\) 不属于 \\(C[0,1]\ftensor C[0,1]\\)。同样的构造给出了 Banach 几乎 \\(f\\)-代数，并对先前发表的关于几乎 \\(f\\)-代数的 Riesz 和 Fremlin 射影张量积的断言产生了影响。

英文摘要

We show that the natural tensor product multiplication of two Archimedean almost \(f\)-algebras does not, in general, extend to their Fremlin tensor product. The two algebras in the counterexample have the same underlying vector lattice \(C[0,1]\). Their multiplications are explicit, positive and nilpotent of index three. The obstruction is the fact that the function \((s,t)\mapsto e^{st}\) does not belong to \(C[0,1]\ftensor C[0,1]\). The same construction gives Banach almost \(f\)-algebras and has consequences for previously published assertions on Riesz and Fremlin projective tensor products of almost \(f\)-algebras.

发表机构

• Research Laboratory of Algebra, Topology, Arithmetic, and Order, Department of Mathematics, Faculty of Mathematical, Physical and Natural Sciences of Tunis, Tunis El Manar University（突尼斯艾曼大学理学院数学系代数、拓扑、算术与序研究实验室）

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