The fundamental theorem of affine geometry in regular L-0-modules

Release time:2024-10-25

Impact Factor:1.3

DOI number:10.1016/j.jmaa.2021.125827

Journal:J. Math. Anal. Appl.

Key Words:Regular L-0-modules L-0-affine mappings Stable mappings The fundamental theorem of affine geometry

Abstract:Let (Omega, F, P) be a probability space and L-0(F) the algebra of equivalence classes of real-valued random variables defined on (Omega, F, P). A left module M over the algebra L-0(F) (briefly, an L-0(F)-module) is said to be regular if x = y for any given two elements x and y in M such that there exists a countable partition {A(n), n is an element of N} of Omega to F such that (I) over tilde (An) . x = (I) over tilde (An ). y for each n is an element of N, where I-An is the characteristic function of A(n) and (I) over tilde (An) its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular L-0 (F)-modules: let V and V' be two regular L-0(F)-modules such that V contains a free L-0(F)-submodule of rank 2, if T : V -> V' is stable and invertible and maps each L-0 -line segment to an L-0-line segment, then T must be L-0-affine.

Co-author:Long Long

First Author:Mingzhi Wu

Indexed by:Journal paper

Correspondence Author:Tiexin Guo

Document Code:125827

Discipline:Natural Science

First-Level Discipline:Mathematics

Document Type:J

Volume:507

ISSN No.:0022-247X

Translation or Not:no

Date of Publication:2021-11-15

Included Journals:SCI

Links to published journals:https://www.sciencedirect.com/science/article/pii/S0022247X21009069