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- Title
Lattice norms on the unitization of a truncated normed Riesz space.
- Authors
Boulabiar, Karim; Hafsi, Hamza
- 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 ⊕ 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 . and we give a necessary and sufficient condition for E ⊕ R to have a lattice norm extending . . Moreover, we show that under this condition, the set of all lattice norms on E ⊕ R extending . has essentially a largest element . 1 and a smallest element . 0 . Also, it turns out that any alternative lattice norm on E ⊕ R is either equivalent to . 1 or equals . 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.
- Subjects
RIESZ spaces; BANACH lattices; DEFINITIONS; AXIOMS
- Publication
Positivity, 2020, Vol 24, Issue 4, p1151
- ISSN
1385-1292
- Publication type
Article
- DOI
10.1007/s11117-019-00722-z