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Semisummands and semiideals in banach spaces

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Abstract

Let |·| be a fixed absolute norm onR 2. We introduce semi-|·|-summands (resp. |·|-summands) as a natural extension of semi-L-summands (resp.L-summands). We prove that the following statements are equivalent. (i) Every semi-|·|-summand is a |·|-summand, (ii) (1, 0) is not a vertex of the closed unit ball ofR 2 with the norm |·|. In particular semi-L p-summands areL p-summands whenever 1<p≦∞. The concept of semi-|·|-ideal (resp. |·|-ideal) is introduced in order to extend the one of semi-M-ideal (resp.M-ideal). The following statements are shown to be equivalent. (i) Every semi-|·|-ideal is a |·|-ideal, (ii) every |·|-ideal is a |·|-summand, (iii) (0, 1) is an extreme point of the closed unit ball ofR 2 with the norm |·|. From semi-|·|-ideals we define semi-|·|-idealoids in the same way as semi-|·|-ideals arise from semi-|·|-summands. Proper semi-|·|-idealoids are those which are neither semi-|·|-summands nor semi-|·|-ideals. We prove that there is a proper semi-|·|-idealoid if and only if (1, 0) is a vertex and (0, 1) is not an extreme point of the closed unit ball ofR 2 with the norm |·|. So there are no proper semi-L p-idealoids. The paper concludes by showing thatw*-closed semi-|·|-idealoids in a dual Banach space are semi-|·|-summands, so no new concept appears by predualization of semi-|·|-idealoids.

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Authors and Affiliations

  1. Departamento de Teoria de Funciones, Facultad de Ciencias, Universidad de Granada, Spain

    J. F. Mena-Jurado, R. Paya-Albert & A. Rodriguez-Palacios

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  1. J. F. Mena-Jurado
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  2. R. Paya-Albert
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  3. A. Rodriguez-Palacios
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Mena-Jurado, J.F., Paya-Albert, R. & Rodriguez-Palacios, A. Semisummands and semiideals in banach spaces. Israel J. Math. 51, 33–67 (1985). https://doi.org/10.1007/BF02772957

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  • DOI: https://doi.org/10.1007/BF02772957

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