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Afak Alpay, Ay§E Uyar on LOCAL APPROXIMATION of POSITIVE OPERATORS 1. Preliminaries If E Is a Riesz Space , the Order Dual Of

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Afak Alpay, Ay§E Uyar on LOCAL APPROXIMATION of POSITIVE OPERATORS 1. Preliminaries If E Is a Riesz Space , the Order Dual Of DEMONSTRATIO MATHEMATICA Vol. XXXV No 3 2002 §afak Alpay, Ay§e Uyar ON LOCAL APPROXIMATION OF POSITIVE OPERATORS 1. Preliminaries If E is a Riesz space , the order dual of E will be the Riesz space of all order bounded linear functionals on E and E~ will denote the order contin- uous functionals on E. Riesz spaces considered in this note are assumed to have separating order duals. If E is a topological Riesz space E' will denote continuous dual of E. Z(E) will denote the ideal centre. In all undefined terminology concerning Riesz spaces we will adhere to the definitions in [1]. DEFINITIONS, a) A Riesz space E is said to have rich centre at Χ G E+ if for each y, 0 < y < χ there exists π 6 Z(E)+ satisfying π(χ) = y. Considering the operator π Λ J, if necessary, we can suppose π satisfies 0 < π < I. E is said to have rich centre if it has rich centre at each χ 6 E+. b) A Riesz space E is said to have topologically full centre at Χ G E+ if the σ(Ε, i?~)-closure of Z(E)x contains the principal order ideal Ix gener- ated by χ. E is said to have topologically full centre if it has topologically full centre at each χ G E+. Every Riesz space that has rich centre at x, also has topologically full centre at that point. Banach lattices with topologically full centre were initiated in [7]. These spaces were also studied in [2], [3] and [6]. The class of Riesz spaces and Ba- nach lattices that have topologically full centre are quite large. σ-Dedekind complete Riesz spaces have topologically full centres. If E is a Banach lattice with a topological orthogonal system then E has topologically full centre. However there are Banach lattices with topologically full centre which are neither σ-Dedekind complete or have a topological orthogonal system. The following two lemmas were inspired by and are similar to Lemma 2.4 in [8]. 1991 Mathematics Subject Classification: 47B65. 608 §. Alpay, A. Uyar LEMMA 1. Consider the following statements on a Riesz space E and χ G E+; a) E has rich centre at x. η b) If 0 < I/I, j/2) ' ' ' > Vn with ^ 2/¿ = χ then there exist 0 < 7r¿ € Z(E) i—l with Σπί = Ie such that π^χ) = yi for each i. c) Ix has Ζ (E)-extension property. Then a <£> b and b c. If, in addition, E is uniformly complete then b&c. Proof, (a => b) The proof will be done by induction. In case η = 1, we have ¡/Ι = ι so we simply take πι = Ie- NOW suppose that the result is known for n+l x η and that we are given 0 < y\, 2/2, · • ·, yn, 2/n+i with ^ 2/fc = - Using the fc=l η inductive hypothesis, we may find 0 < 7ΓΙ,7Γ2, · · · , πη with ^ π* = I e and i=1 7Ti(x) = yi and πη(χ) = yn+yn+i· As 0 < yn < x, there exists π G Z(E) with 0 < 7Γ < I satisfying π(χ) — yn. Let Rn = πΛπη, Rn+i = πη — (πΛιrn). Then 0 < Rn, Rn+i < I and Rn + Rn+i = 7Tn. So letting Ri = 7r¿ for 1 < i < η — 1 n+l η we have Ri = = I, Rn(x) = 7/n, Än+i(x) = 2/n+i as asserted. í=l ¿=1 (6 =í> o) For 0 < y < χ. Let yi = χ — y, 2/2 = y, we can find πι,π2 G Ζ (E) with πι + 7Γ2 = I and π*(χ) = yi. As π2(χ) = y, E has rich centre at x G £+. (ò => c) Let π G Ζ(ΙΧ) with 0 < π < I. As 0 < π(χ) < χ. We can find R G Ζ (E) with R(x) = π(χ). But then R = π on Jx. In presence of uniform completeness, Ix is a unital /-algebra which have rich centre at their units. • DEFINITION. An ideal I in a Banach lattice E is said to have the approximate extension property if for each π in Z(I) there exists (πη) in Z(E) such that πη(χ) —» π(χ) for each x G The proof of the next lemma is similar to the preceding and will be omitted. LEMMA 2. For a Banach lattice E and x G E+, the following are equivalent statements; a) E has topologicaly full centre at x. η are x b) If e > 0 and 0 < 2/1, · · · ,yn given with ^ t/i = > then there exist η i—l 0 < TTi G Z(E) with ^iTi = I and ||πί(χ) — yi\\ < e for each i. i=l Approximation of positive operators 609 c) Ix has approximate Ζ (E)-extension property, m LEMMA 3. Let E be a Banach lattice with a quasi-interior point. Then E has topologically full centre. Proof. It suffices to show that Ix, χ e E+, has the approximate extension property [3]. That is, for each π0 e Z(IX) there exist (πη) C Z(E) such that TTn(y) Tro(y) for each y in Ix. As the set of quasi-interior points is dense in E+, there is a sequence (un) of quasi-interior points in E+ converging to 7To(x). Let In be the principal order ideal generated by un V χ and let πη : In —> In be the multiplication operator defined as πn(y) = un-y. irns extend to operators πη in Z(E) and 7Tn(x) = 7tn(x) = χ • un = (x A un)(x V un) = χ A un is satisfied for each n. Continuity of the lattice operations in E imply that un Α χ —• 7Γο(χ) Λ χ = 7Γο(ι) and Πη(χ) —• TTQ(X). Let 0 < y < χ be arbitrary, then l^n(y) - 7Ì"o(ì/)| < ITT„ - 7To|x < |πη(χ) - 7Γ0(χ)| shows that Ix has the approximate extension property and the proof is complete. • A Riesz space E has locally rich centre at each of its quasi-interior points (if any). But an element that has locally rich centre need not be a quasi- interior point. For example, the Dedekind complete Riesz space Í1 has locally rich centre at β2 — (0,1,0,0, · · ·) but ei is not a quasi-interior point. Let now E, F be Riesz spaces with F Dedekind complete. For every π e Ζ (E) and σ Ε Ζ (F), the mapping Τ -> σΤπ, Τ Ε Lb(E, F) is an element of Z(Lb(E,F)). We denote this element by σ (8) ir. This mapping is rarely onto [8]. We define Z{F) ® Z(E) to be the linear subspace of Z(Lb(E,F)) generated by elements of the form σ ® π with π e Ζ (Ε), σ € Ζ {F). If V denotes the band projections of F, V®Z(E) is the subspace of Z(F)®Z(E) generated by elements of the form Ρ <g> π, Ρ € V and π G Ζ (E). Every element of V ® Z(E) has a unique representation as a finite sum of the form η Σ Pi® 7Ti, Pi e V and tt¿ G Ζ (E), 1 <i<n such that P¿ A Pj = 0 if i φ j, ι=1 η Τ: Pi = IF and π* φ χ, if i φ j. This representation will be referred to i=l as standart representation. V ® Z(E) is a Riesz subspace of Z(Lb(E,F)). In general, Z(F) ® Z(E) is not a Riesz subspace of Z(Lb(E,F)). We refer the reader to [4] for details. If 0 < <¿> € F~, 0 < Τ G Lb(E,F) and 0 < χ β E, define Φ„ιΓ|Χ : Z(Lb(E,F)) —• R by setting Φν,τ,χ(π) = φ{π(Τ)χ), 610 Alpay, A. Uyar π G Z(Lb(E,F)). Define \$Ψ,τ,χ • 0 < φ G F~,0 < Γ G Lb(E,F),0 < χ G eJ Τ separates the points of Z(Lb(E,F)) if separates the points of F. If we define + ε = (v® ζ (Ε)) Η [O,If®Ie}. The closure of £+ for \a\{Z(Lb(E,F),F) is |||T £+ and we have the fol- lowing result which was first proved for E — C(K) and Dedekind complete Riesz space F in [4]. LEMMA 4. Let E be a Riesz space with rich centre and F be Dedekind com- + plete Riesz space. Then £ is \a\(Z(Lb(E, F),F) dense in [0, lLb(E,F)\· Proof. Let e > 0, 0 < ψ e 0 < Τ G Lb(E,F), 0 < χ G E and ϋ G [0, J^^^)] be given. We show there exists μ G £+ such that — μ\) < e. Assume first that ϋ is a principal component of lLb(E,F)· Then S = ΰ(Τ) is a principal component of Τ in Lb(E, F). Since S A (T — S) = 0, it follows from Lemma 2.3 in [4] that there exist disjoint band projections = Ph ' ' ' ) Pm ^ Ρ with P\ -(-•·· Pm I ρ and elements 0 ^ x\, · · ·, xn in E with x\ + · · · + xn = χ such that Σ < PjSxi, φ > A < Pj{T - S)xi, Ψ><\· Define Q = {(i,j) :< PjSxi, ψ > < Pj{T - S)xi,φ >} and let Q' = {1,2, · · ·, η} χ {1,2, · · ·, m) \ Q. It follows that Σ < PjSxi, φ > <| and £ < Pj{T - S)xh ψ > < (i,j)eQ (i,j)eQ' There exist 0 < πι, · · ·, 7rn G Ζ (E) such that -κ\ + · · · + 7rn = Ie and Xi = 7T¿ (χ), 1 < i < η by Lemma 1 above.
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