Directed Ideals in partially ordered vector spaces

Transcription

1 Directed Ideals in partially ordered vector spaces Anke Kalauch TU Dresden Leiden, July 2013 Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

2 Outline 1 Characterizations of ideals in Riesz spaces 2 Full subspaces 3 Solid and solvex subspaces 4 Kernels of homomorphisms Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

3 Characterizations of ideals in Riesz spaces Motivation: Ideals in Riesz spaces Let X be a Riesz space and let I be a linear subspace of X. Each of the following properties of I is equivalent to I being an ideal in X: (a) I is a Riesz subspace of X and for every x X, y, z I with y x z one has x I. (b) For every x X, y I with x y one has x I. (c) For every y I, the convex solid hull of {y} in X is contained in I. (d) I is a Riesz subspace of X and there exist a Riesz space Y and a positive linear map T : X Y such that I = T 1 ({0}). (e) There exist a Riesz space Y and a Riesz homomorphism T : X Y such that I = T 1 ({0}). Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

4 Characterizations of ideals in Riesz spaces Aim generalizations of (a) (e) to partially ordered vector spaces investigation of directed subspaces Joint work with Onno van Gaans, Universiteit Leiden Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

5 Characterizations of ideals in Riesz spaces Basic notions Let X be a vector space and an partial order on X such that (i) x, y, z X with x y x + z y + z (ii) x 0, λ [0, ) λx 0 Cone of all positive elements: K := {x X : x 0} We call (X, K ) a partially ordered vector space (povs). Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

6 Full subspaces Full subspaces Let (X, K ) be a povs and I a subspace of X. The following three properties of I are equivalent: (a1) I is a full subspace, i.e. for every y, z I the order interval [y, z] := {x X : y x z} is contained in I. (a2) For every y I and x X with y x y one has x I. (a3) For every y I and x X with 0 x y one has x I. Definitions of ideal in povs: [Kadison 1951], [Bonsall 1954] (a2) [Fuchs 1965] directed subspace with (a1) Every subspace without nontrivial positive elements is full. I is a full subspace if and only if (I K ) (I K ) is a full subspace. Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

7 Solid and solvex subspaces The modul x is replaced by {x, x} u Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

8 Solid and solvex subspaces Solid subspaces Idea: In the definition of an ideal (= solid subspace) in Riesz spaces the modulus x := x ( x) is used; in povs it is replaced by the set {x, x} u of upper bounds of x and x. Definition (v.g. 2003) A subspace I of a povs X is called solid if for every y I and x X with { x, x} u { y, y} u one has x I. By the same method, in [K.,v.G. 2006] disjointness is introduced in povs by defining x y {x + y, x y} u = {x y, x + y} u. Bands are defined to be sets that equal their double-disjoint complement. In Archimedean directed povs, every band is a solid subspace. In contrast to full subspaces, a subspace without nontrivial positive elements need not be solid. Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

9 Solid and solvex subspaces Properties of solid subspaces Proposition (K., v.g. 2013) If I is solid then I is full. If I is full and directed, then I is solid. If I is solid then (I K ) (I K ) is solid. Example: X := R 3, K := pos{(1, 0, 1), (0, 1, 1), ( 1, 0, 1), (0, 1, 1)} x 0 := (1, 0, 0), x 1 = (0, 1, 0), x 2 = (0, 0, 1), x 3 = (1, 1, 0) There exist full subspaces that are not solid. Indeed, I 0 := span{x 0 } is full, but not solid, since { x 0, x 0 } u = K + (0, 0, 1) = { x i, x i } u, i = 1, 2, and if I is a solid subspace containing x 0, then I = R 3. A subspace with a solid directed part need not be solid. Indeed, the directed part of I 0 equals {0}, which is solid. There exist solid subspaces that are not directed. Indeed, span{x 3 } is a band and therefore a solid subspace, which is not directed. Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

10 Solid and solvex subspaces Solvex subspaces: Motivation If X is an Archimedean directed povs, then there is a Riesz space V and a bipositive linear map i : X V such that i[x] is order dense in V, i.e. y V : y = inf{i(x); x X, i(x) y}. (V, i) is called a vector lattice cover of X. Even more general: See pre-riesz spaces [van Haandel 1993] The restriction of an ideal in V is a solid subspace in X. There are solid subspaces in X that are not obtained in this way. We characterize all subspaces of X that are restrictions of ideals in V. Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

11 Solid and solvex subspaces Solvex subspaces Definition (v.g. 2003) A set A X is called solvex if for every x X, x 1,..., x n A and λ 1,..., λ n (0, 1] with n k=1 λ k = 1 such that { n } u { x, x} u ε k λ k x k : ε 1,..., ε n { 1, 1} one has that x A. k=1 If X is a Riesz space, then A is solvex if and only if A is solid and convex. Proposition (K., v.g. 2008) Let (V, i) be a vector lattice cover of the povs X. A subspace I of X is solvex if and only if there exists an ideal J in V such that I = {x X : i(x) J}. Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

12 Solid and solvex subspaces Ideals and Bands in pre-riesz spaces Archimedean vector lattice pre-riesz space band = o-closed ideal band o-closed ideal solvex ideal ideal = solvex ideal ideal Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

13 Kernels of homomorphisms Kernels of homomorphisms Definition (van Haandel, 1993) Let (X, K ) and (Y, L) be povs and T : X Y a linear operator. T is called a - Riesz homomorphism if for every x, y X one has T ({x, y} u ) l = {T (x), T (y)} ul, - Riesz* homomorphism if for every x, y X one has T ({x, y} ul ) {T (x), T (y)} ul. Every Riesz homomorphism is a Riesz* homomorphism, and every Riesz* homomorphism is positive. If X and Y are Riesz spaces, then the notions Riesz homomorphism and Riesz* homomorphism coincide. Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

14 Kernels of homomorphisms Directed kernels The composition of two Riesz homomorphisms need not be a Riesz homomorphism. The composition of two Riesz* homomorphisms is again a Riesz* homomorphism. In a povs the kernel of a Riesz homomorphism is not directed, in general (see e.g. circular cones). Proposition (K., v.g. 2013) Let (X, K ) and (Y, L) be povs and let T : X Y be a positive linear operator such that for each w L there is x K with T (x) = w. If the kernel T 1 ({0}) is directed, then T is a Riesz homomorphism. Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

15 Kernels of homomorphisms Quotient maps Let (X, K ) be a povs and I a linear subspace of X. For x X denote [x] = {y X : y x I}. Denote ˆX = X/I = {[x]: x X} and q : X ˆX, x [x] (quotient map). The set ˆK = {[x]: x K } is a wedge in ˆX. If I is, in addition, full, then ˆK is a cone in ˆX. Proposition (K., v.g. 2013) If I is a directed full subspace of X, then the quotient map q : X X/I is a Riesz homomorphism. Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

16 Kernels of homomorphisms Kernels of positive maps and Riesz* homomorphisms Proposition (K., v.g. 2013) A subspace I of a povs X is full if and only if there exist a partially ordered vector space (Y, L) and a positive linear map T : X Y such that I = T 1 ({0}). Proposition (K., v.g. 2013) Let (X, K ) and (Y, L) be partially ordered vector spaces. If T : X Y is a Riesz* homomorphism, then the kernel T 1 ({0}) is a solvex subspace of X. Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

17 Kernels of homomorphisms Directed ideals in povs Theorem (K., v.g. 2013) Let (X, K ) be a partially ordered vector space and let I be a directed subspace of X. The following six properties of I are equivalent: (i) I is full. (ii) I is solid. (iii) I is solvex. (iv) There exist a partially ordered vector space (Y, L) and a positive linear map T : X Y such that I = T 1 ({0}). (v) There exist a partially ordered vector space (Y, L) and a Riesz* homomorphism T : X Y such that I = T 1 ({0}). (vi) There exist a partially ordered vector space (Y, L) and a Riesz homomorphism T : X Y such that I = T 1 ({0}). Proof: (iii) = (ii) (i) (iv) (i) = (vi) (quotient map), (v) = (iii) Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

18 Conclusion Kernels of homomorphisms We know the relation of bands and solid and solvex subspaces in pre-riesz spaces; properties of directed solid subspaces as in the main theorem. We do not know whether similar properties are satisfied for non-directed solid subspaces; characterization/properties of directed bands. Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

19 References Kernels of homomorphisms O. van Gaans, A. Kalauch Directed ideals in partially ordered vector spaces to appear in Indag. Math. A. Kalauch, B. Lemmens, O. van Gaans Riesz completions, functional representations and anti-lattices to appear in Positivity O. van Gaans, A. Kalauch Ideals and bands in pre-riesz spaces Positivity 12(2008), no. 4, Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20

20 Kernels of homomorphisms Thank you for your attention! Anke Kalauch (TU Dresden) Directed ideals Leiden, July / 20