Truncated Riesz spaces was first introduced by Fremlin in the context of real-valued functions. An appropriate axiomatization of the concept was given by Ball. Keeping only the first Ball’s Axiom (among three) as a definition of truncated Riesz spaces, the first named author and El Adeb proved that if E is truncated Riesz space then \(E\oplus \mathbb {R}\) can be equipped with a non-standard structure of Riesz space such that E becomes a Riesz subspace of \(E\oplus \mathbb {R}\) and the truncation of E is provided by meet with 1. In the present paper, we assume that the truncated Riesz space E has a lattice norm \(\left\| .\right\| \) and we give a necessary and sufficient condition for \(E\oplus \mathbb {R}\) to have a lattice norm extending \(\left\| .\right\| \). Moreover, we show that under this condition, the set of all lattice norms on \(E\oplus \mathbb {R}\) extending \(\left\| .\right\| \) has essentially a largest element \(\left\| .\right\| _{1}\) and a smallest element \(\left\| .\right\| _{0}\). Also, it turns out that any alternative lattice norm on \(E\oplus \mathbb {R}\) is either equivalent to \(\left\| .\right\| _{1}\) or equals \(\left\| .\right\| _{0}\). As consequences, we show that \(E\oplus \mathbb {R}\) is a Banach lattice if and only if E is a Banach lattice and we get a representation’s theorem sustained by the celebrate Kakutani’s Representation Theorem.

中文翻译：

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截短赋范Riesz空间的单位化的格范规范

截断的Riesz空间最早是由Fremlin在实值函数的上下文中引入的。鲍尔对这个概念进行了适当的公理化。仅保留第一个Ball的公理（其中三个）作为截尾的Riesz空间的定义，第一名作者和El Adeb证明，如果E是截尾的Riesz空间，则\（E \ oplus \ mathbb {R} \）可以配备Riesz空间的非标准结构，使得E变成\（E \ oplus \ mathbb {R} \）的Riesz子空间，并且E的截断由满足1提供。在本文中，我们假定截断了里兹空间E具有格范数\（\ left \ | .. \ right \ | \）并为\（E \ oplus \ mathbb {R} \）给出一个扩展\（\ left \ |。\ right \ | \）的格范数提供了充要条件。此外，我们表明在这种情况下，在\（E \ oplus \ mathbb {R} \）上扩展\（\ left \ |。\ right \ | \）上的所有晶格范数的集合实际上具有最大的元素\（\ left \ |。\ right \ | _ {1} \）和最小元素\（\ left \ |。\ right \ | _ {0} \）。而且，事实证明，\（E \ oplus \ mathbb {R} \）上的任何替代格范数都等于\（\ left \ |。\ right \ | _ {1} \）或等于\（\ left \ |。\ right \ | _ {0} \）。结果表明，\（E \ oplus \ mathbb {R} \）是一个Banach格，并且仅当E是一个Banach格并且我们得到了庆祝的角谷的表示定理所支持的表示定理。