A note on the sum of closed ideals and Riesz subspaces

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• 作者：Witold Wnuk (1)
  
  1. Faculty of Mathematics and Computer Science ; A.聽 Mickiewicz University ; Umultowska 87 ; 61鈥?14聽 ; Poznan ; Poland
  
• 关键词：Continuous Riesz space ; Discrete Riesz space ; Banach lattice ; Order continuous norm ; 46B42 ; 46A40 ; 46B45
• 刊名：Positivity
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：19
• 期：1
• 页码：137-147
• 全文大小：204 KB
• 参考文献：1. Aliprantis, C., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, 2nd edn, vol 105. American Mathematical Society, Providence, RI (2003)
  2. Aliprantis, C, Burkinshaw, O (2006) Positive Operators. Springer, Berlin CrossRef
  3. Davis, EB (1968) The structure and ideal theory of the predual of a Banach lattice. Trans. Am. Math. Soc. 131: pp. 544-555 CrossRef
  4. Drewnowski, L.: On minimal topological linear spaces and strictly singular operators. Comment. Math. Tomus Specialis II, 89鈥?06 (1979)
  5. Drewnowski, L (1984) Quasi-complements in F-spaces. Studia Math. 77: pp. 373-391
  6. Drewnowski, L (2011) On infinite sums of closed ideals in F-lattices. Func. Approx. Comment. Math. 44: pp. 279-284 CrossRef
  7. Lindenstrauss, J, Tzafriri, L (1979) Classical Banach Spaces II. Function Spaces. Springer, Berlin, New York
  8. Lotz, HP (1968) 脺ber das Spektrum positiver Operatoren. Math. Z. 108: pp. 15-32 CrossRef
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  10. Luxemburg, WAJ, Zaanen, AC (1971) Riesz Spaces I. North-Holland, Amsterdam
  11. Mackey, GW (1945) On infinite dimensional linear spaces. Trans. Am. Math. Soc. 57: pp. 155-207 CrossRef
  12. Meyer-Nieberg, P (1991) Banach Lattices. Springer, Berlin Heidelberg New York
  13. Wilansky, A (1975) Semi-Fredholm maps of FK-spaces. Math. Z. 144: pp. 9-12 CrossRef
  14. Wnuk, W (1986) When is the closure of an atomic Riesz subspace atomic?. Bull. Acad. Pol. Sci. Math. 34: pp. 659-694
  15. Wnuk, W (1999) Banach Lattices with Order Continuous Norms, Advanced Topics in Mathematics. Polish Scientific Publishers PWN, Warsaw
  16. Wnuk, W., Wiatrowski, B.: On the Levi and Lebesgue properties. Indag. Math. (N.S.) 18(4), 641鈥?50 (2007)
  17. Wnuk, W.: Some remarks on the algebraic sum of ideals and Riesz subspaces. Canad. Math. Bull. 56(2), 434鈥?41 (2013). doi:10.4153/CMB-2011-151-0
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Operator Theory
  Potential Theory
  Calculus of Variations and Optimal Control
  Econometrics
  
• 出版者：Birkh盲user Basel
• ISSN：1572-9281

文摘

We indicate a broad class of closed ideals \(F\) in a Banach lattice \(E\) such that there exists a closed continuous (and also a closed heterogeneous) Riesz subspace \(G \subset E\) satisfying the following two conditions: \(G \cap F = \{0\}\) and the algebraic sum \(G + F\) is not closed.