Lattice norms on the unitization of a truncated normed Riesz space

Abstract

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.