Positively Convex Modules and Ordered Normed Linear Spaces
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Journal of Convex Analysis Volume 10 (2003), No. 1, 109–127 Positively Convex Modules and Ordered Normed Linear Spaces Dieter Pumpl¨un Fachbereich Mathematik, FernUniversit¨at,58084 Hagen, Germany [email protected] Dedicated to Harald Holmann on the occasion of his 70th birthday Received October 11, 2001 A positively convex module is a non-empty set closed under positively convex combinations but not necessarily a subset of a linear space. Positively convex modules are a natural generalization of positively convex subsets of linear spaces. Any positively convex module has a canonical semimetric and there is a universal positively aáne mapping into a regularly ordered normed linear space and a universal completion. 2000 Mathematics Subject Classiﬁcation: 52A05, 52A01, 46B40, 46A55 1. Introduction In the following all linear spaces will be real. If E is a linear space ordered by a proper cone C; E = C C, and normed by a Riesz norm with respect to C, the positive part of the unit ball− O(E), i.e. O(E) C, is a typical example of a positively convex set or module. Actually any positively\ convex set or module i.e. any non-empty set P closed under positively convex operations, is an aáne quotient of such a positive part of a unit ball. Positively convex modules are natural generalizations of positively convex sets, the postulate that they are subsets of some linear space is dropped. Special positively convex sets play an important part in the theory of order-unit Banach spaces as so-called “universal capsÔ (cp. [21], §9) and also as generators of the topology of locally solid ordered topological linear spaces and the theory of Riesz seminorms (cp. [21], §6). In [9] it is proved that there is a functorial connection between the theory of positively (countably) convex sets or modules and the theory of regularly ordered Banach spaces, i.e. ordered by a Riesz norm. To investigate this relation and to give a more explicit description Wickenh¨auserin [20] introduced a semimetric for positively convex modules. In [4] Kemper expressed this semimetric by the seminorm of the positively convex module. These results improved the description of the universal regularly ordered normed linear space generated by a positively convex module somewhat but the situation has been far from satisfactory compared with the results for other types of convex sets (cf. [1], [10], [11], [12], [16], [14], [15]). In §2 of this paper the semimetric of a positively convex module is introduced and dis- cussed. §3 contains the functorial connection between positively convex modules and ordered linear spaces and the characterization of preseparated positively convex modules. ISSN 0944-6532 / $ 2.50 c Heldermann Verlag 110 D. Pumpl¨un/ Positively Convex Modules and Ordered Normed Linear Spaces The positive part of the unit ball of a regularly ordered normed linear space induces a canonical functor to the category of positively convex modules. In §4 it is shown that, vice versa, any positively convex module generates a universal regularly ordered normed linear space and a universal positively aáne morphism into the positive part of the unit ball of this space. This leads to a discussion of complete positively convex modules and positively superconvex modules in §5 where also the existence of a universal completion is proved. 2. The semimetric In the following all linear spaces considered will be real. A positively convex set X in a linear space E is a non-empty subset of E closed under positively convex operations i.e. n n xi X; 1 i n, and «i 0; «i 1 implies «ixi X.A positively convex 2 ´ ´ ≥ i=1 ´ i=1 2 operation may be described as an elementP of the set ΩPpc := «Ý «Ý = («1;:::;«n); n n f j 2 N;«i 0; 1 i n, and «i 1 . This leads to the following natural generalization. ≥ ´ ´ i=1 ´ g P Deﬁnition 2.1. (cf. [4], [5], [9], [13], [16]): A positively convex module P is a non-empty n set together with a family of mappings« ÝP : P P; «Ý = («1;:::;«n) Ωpc. In addition, with the notation −! 2 n «ipi :=« ÝP (p1; : : : ; pn) i=1 X for p P; 1 i n, the following equations have to be satisﬁed: i 2 ´ ´ n (PC1) ®ikpi = pk; i=1 X p P; 1 i n, and ® the Kronecker symbol, 1 i; k n. i 2 ´ ´ ik ´ ´ n m n (PC2) «i ¬ikpk = 0 «i¬ik1 pk; i=1 Àk Ki ! k=1 i=1 X X2 X kXKi B 2 C @ A n where («1;:::;«n); (¬ik k Ki) Ωpc; 1 i n; pk P , for k Ki. Moreover, j 2 2 ´ ´ 2 2 i=1 n S Ki = Nm = k 1 k m and in the “sumÔ ¬ikpk the summands are supposed i=1 f j ´ ´ g k K 2 i toS be written in the natural oder of the k's. P A number of computational rules follow from these equations (cf. [9], [10]), e.g. the fact n that «ipi is not changed by adding or omitting summands with zero coeácients. Hence, i=1 (PC2)P takes the more simple form n m m n «i ¬ikpk = «i¬ik pk: i=1 À ! À i=1 ! X Xk=1 Xk=1 X D. Pumpl¨un/ Positively Convex Modules and Ordered Normed Linear Spaces 111 Obviously, any convex set, which will always mean a positively convex subset in some linear space, is a positively convex module. Any positively convex module is a convex module (cp. [16]), the converse does not hold, because any positively convex module n contains a zero element 0 := 0pi. However, every absolutely convex module ([10], [14]) i=1 is a positively convex module,P as is any cone or convex set in a linear space containing the orgin. A familiar example of positively convex sets are the universal caps of cones in functional analysis ([21]). If P1;P2 are positively convex modules a mapping f : P1 P2 is called a morphism or a positively aáne mapping if −! n n f «ixi = «if(xi) À i=1 ! i=1 X X holds for any (« ;:::;« ) Ω and any x P; 1 i n. This deﬁnes the category 1 n 2 pc i 2 ´ ´ PC of positively convex modules with forgetful functor U : PC Set. For any X (X) ! 2 Set; ∆(R+ ) := h h : X R+, support of h ﬁnite and h(x) 1 ; R+ := f j −! x ´ g t t R; t 0 , is the free positively convex module generatedP by X. ([9], [10]). f j 2 ≥ g + U: PC Set is algebraic. It is the Eilenberg-Moore category of the category Vec1 of −! + regularly ordered normed linear spaces where ∆ : Vec1 Set is the forgetful functor with ∆(E) := x x E; x 0; x 1 ([4], [9]). −! f j 2 ≥ k k ´ g Deﬁnition 2.2. (cf. [4], [20]): For a positively convex module P and x; y P one deﬁnes 2 d (x; y) := inf « ¬ > 0; « > 0 and there are a; u; v P and P f 2¬ j 2 ®; " R+ with «+¬; ®+¬; "+¬ 1 and «a+¬x = ®u+¬y; «a+¬y = "v+¬x . 2 ´ g dP is called the semimetric of P (cf. 2.3). If it is obvious to which P the semimetric belongs the index is often omitted. P PC is called a metric positively convex module 2 if dP is a metric. This semimetric was ﬁrst introduced by Wickenh¨auserin [20], 9.2 in the following form dW (x; y) = inf « 0 « 2 and there are a; u; v P and n N n 3 with 1 xf+ j« a ´= 1 y´+ 1 v; 1 y + « a = 1 x + 12u . 2 ≥ n n n 2 n n n 2 g This form proved to be not very well suited for computations. A straightforward argument shows that 1 d (x; y) = d (x; y): P 2 W In [4], Kemper expressed dW (x; y) by the seminorm of positively convex modules (cf. [9]), a form which still presented considerable diáculties in describing the so-called comparison + functor ([9]) from PC to Vec1 . This functor will be investigated in §4. Proposition 2.3. ([4], [16], [20]): (i) For any positively convex module P , dP is a semimetric with 0 dP (x; y) 1 and d (x; 0) x ; x; y P , where denotes the seminorm of P´([9]). ´ P ´ k k 2 k£k 112 D. Pumpl¨un/ Positively Convex Modules and Ordered Normed Linear Spaces (ii) Let x ; y P; 1 i n; P PC and (λ ; : : : ; λ ) Ω then i i 2 ´ ´ 2 1 n 2 pc n n n d λ x ; λ y λ d (x ; y ): P i i i i ´ i P i i À i=1 i=1 ! i=1 X X X (iii) for x; y P; P PC and 0 λ 1 2 2 ´ ´ dP (λx, λy) = λdP (x; y): (iv) If f : P P is a morphism in PC then 1 −! 2 d (f(x); f(y)) d (x; y); 2 ´ 1 if di(£; £) is the semimetric of Pi; i = 1; 2. Proof. (i): 2.2 implies that dP (x; y) is symmetric. In [20] 0 dP (x; y) 1 was shown. 1 1 1 1 1 ´ ´ One takes « = 2− ; ¬ = 4− ; ® = " = 2− ; a = 2− x + 2− y; u = x and v = y. In order to show the triangle inequality for x; y; z P , we may assume d (x; z) + d (z; y) < 1. 2 P P Hence, dP (x; z) < 1 and dP (z; y) < 1 follows and there are equations «1a1 + ¬1x = ®1u1 + ¬1z «1a1 + ¬1z = "1v1 + ¬1x «2a2 + ¬2z = ®2u2 + ¬2y «2a2 + ¬2y = "2v2 + ¬2z 1 1 with dP (x; z) «1(2¬1)− < 1; dP (z; y) «2(2¬2)− < 1.