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On self-majorizing elements in Archimedean vector lattices

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Abstract

A finite element in an Archimedean vector lattice is called self-majorizing if its modulus is a majorant. Such elements exist in many vector lattices and naturally occur in different contexts. They are also known as semi-order units as the modulus of a self-majorizing element is an order unit in the band generated by the element. In this paper the properties of self-majorizing elements are studied systematically, and the relations between the sets of finite, totally finite and self-majorizing elements of a vector lattice are provided. In a Banach lattice an element \(\varphi \) is self-majorizing , if and only if the ideal and the band both generated by \(\varphi \) coincide.

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Notes

  1. The formulation of Theorem 1 in [6] has to be corrected: \(E\) and \(F\) have to be Banach lattices.

  2. And to avoid unsuitable vector lattices with no proper maximal ideals at all, see e.g., [11], Example 27.8.

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Authors and Affiliations

  1. Fraunhofer Institut for Industrial Mathematics ITWM, Kaiserslautern, Germany

    Katrin Teichert

  2. Fachrichtung Mathematik, Technische Universitaet Dresden, Dresden, Germany

    Martin R. Weber

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  1. Katrin Teichert
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  2. Martin R. Weber
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Correspondence to Martin R. Weber.

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Teichert, K., Weber, M.R. On self-majorizing elements in Archimedean vector lattices. Positivity 18, 823–837 (2014). https://doi.org/10.1007/s11117-014-0278-4

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