It is proved that, for a pre-ordered vector space $X$, the quotient space $(X/A,[W])$ is the Archimedeanization of $X$, where $W$ is the closure of the positive wedge $X_+$ in the ru-topology, $A=W\cap(-W)$, and $[W]$ is the quotient set of $W$ in $X/A$.
Archimedean property is important in the functional representation of various analytic and algebraic structures. For example, Kadison's representation theorem [7] tells that every ordered real vector space with an Archimedean order unit is order isomorphic to a vector subspace of the space of continuous real valued functions on a compact Hausdorff space. Kadison's theorem inspired M.D. Choi and E.G. Effros to obtain an analogous representation theorem for selfadjoint subspaces of unital C * -algebras that contain the unit [2].
There is a plenty of recent works by many mathematicians who studied the Archimedean property in connection to various applications in algebra, analysis and quantum physics. In the breakthrough paper [12], V.I. Paulsen and M. Tomforde have developed a theory of ordered *-vector spaces. In their theory, Archimedeanization of ordered vector spaces with order units plays a crucial role. The construction of Archimedeanization of ordered vector spaces with an order unit [12] was extended to arbitrary ordered vector spaces in [3]. We also mention one more important direction in analysis (based on representation theorems for Archimedian structures), the construction of free analytic spaces that was initiated by B. de Pagter and A.W. Wickstead in their pioneered work [11].
The present paper is devoted to an interplay between the Archimedean property and relatively uniform convergence in (pre-)ordered vector spaces. In Theorem 3.3, we prove that the quotient set [W ] of the τ ru -closure of the positive wedge X + of a pre-ordered vector space X is an Archimedean cone in quotient space X/(W ∩ (-W )). The main result of our paper is Theorem 3.5 which establishes that (X/(W ∩ (-W )), [W ]) is the Archimedeanization of X.
In what follows, vector spaces are real and operators are linear. We recall few well known definition. For further unexplained terminology and notations we refer to [1,5,8].
A subset W of a vector space X is a wedge if W + W ⊆ W and tW ⊆ W for t ⩾ 0. A wedge C is a cone whenever C ∩ (-C) = {0}. Under an ordered vector space (briefly, OVS) we understand a vector space X together with a cone (denoted by X + and called a positive cone of X). Replacing a cone by a wedge, we also consider a pre-ordered vector space (briefly, POVS). A wedge W in a vector space X is majorizing (or generating) if
Every OVS (POVS) X is equipped with a partial (pre-)order
x ⩽ y ⇐⇒ y -x ∈ X + .
Every pair of vectors {a, b} in an OVS (POVS) X produces an order interval (possibly empty,
A subset A of a POVS is called an order ideal in X, whenever A is a vector subspace of
Clearly, each almost Archimedean POVS must be an OVS. Moreover, every Archimedean OVS is almost Archimedean. The converse is not true even for two-dimensional OVSs (see Example 2.5 in [3]). It is also straightforward to see: a POVS X is almost Archimedean iff X + does not contain a straight line.
A net (x α ) in a POVS X ru-converges to x ∈ X (shortly, x α ru -→ x) if, for some (a regulator of the convergence) w ∈ X + , there exists a sequence (α n ) of indices with x α -x ⩽ ± 1 n w for α ⩾ α n . Whenever we need to specify a regulator w, we write x α ru -→ x(w).
Clearly, every ru-convergent net in an OVS X has a unique ru-limit iff X is almost Archimedean. As the ru-convergence is sequential, we can always restrict ourselves to ru-convergent sequences. The ru-convergence is an abstraction of the classical uniform convergence of functions in C[0, 1]. The notion of ru-convergence was introduced by L.V. Kantorovich [9].
3 Closedness of the positive wedge in ru-topology vs Archimedean property in pre-ordered vector spaces
The following lemma is a standard fact. Lemma 3.1. Let W be a wedge in a vector space X. Then [W ] is a cone in X/A, where
An OVS X is a vector lattice if x + := inf X + ∩ (x + X + ) exists for every x ∈ X. It is easy to see that every almost Archimedean vector lattice is Archimedean. An order ideal A in a vector lattice X will be referred to as a lattice ideal whenever it satisfies the property:
W.A.J. Luxemburg and L.C-Jr. Moore proved in [10] that the ru-closed subsets of a vector lattice X are exactly the closed sets of the so-called ru-topology on X. This topology can be also introduced on an arbitrary POVS (cf., [8]).
Definition 3.1. The ru-topology, denoted by τ ru , on a POVS X is determined as follows: a subset S of X is τ ru -closed whenever S ∋ x n ru -→ x implies x ∈ S. The τ ru -closure of a subset S of X is denoted by S ru .
It follows immediately from Definition 3.1 that the ru-topology on a POVS X is the strongest topology τ on X with the property
The ru-topology is not linear in general. It is proved in [10,Theorem 4.2] that every lattice homomorphism between vector lattices is continuous in the ru-topology. Indeed, it is true for arbitrary positive operators between POVSs, and if additionally the positive wedge in the co-domain is majorizing, it is also true for every order bounded operator.
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