2019/10/25 by Boulabiar, Karim, Hafsi, Hamza
#46A40 #46B40 #46B42 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1910.11715
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⊕ℝ can be equipped with a non-standard structure of Riesz space such that E becomes a Riesz subspace of E⊕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 \Vert .\Vert and we give a necessary and sufficient condition for E⊕ℝ to have a lattice norm extending \Vert .\Vert . Moreover, we show that under this condition, the set of all lattice norms on E⊕ℝ extending \Vert .\Vert has essentially a largest element \Vert .\Vert 1 and a smallest element \Vert .\Vert 0. Also, it turns out that any alternative lattice norm on E⊕ℝ is either equivalent to \Vert .\Vert 1 or equals \Vert .\Vert 0. As consequences, we show that E⊕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.