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The ℒ p spaces

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Abstract

The ℒ p spaces which were introduced by A. Pełczyński and the first named author are studied. It is proved, e.g., that (i)X is an ℒ p space if and only ifX* is and ℒ q space (p −1+q −1=1). (ii) A complemented subspace of an ℒ p space is either an ℒ p or an ℒ2 space. (iii) The ℒ p spaces have sufficiently many Boolean algebras of projections. These results are applied to show thatX is an ℒ∞ (resp. ℒ1) space if and only ifX admits extensions (resp. liftings) of compact operators havingX as a domain or range space. We also prove a theorem on the “local reflexivity” of an arbitrary Banach space.

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Authors and Affiliations

  1. Hebrew University of Jerusalem, Israel

    J. Lindenstrauss & H. P. Rosenthal

  2. University of California, Berkeley

    J. Lindenstrauss & H. P. Rosenthal

Authors
  1. J. Lindenstrauss
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  2. H. P. Rosenthal
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Additional information

This research was partially supported by NSF Grant# 8964.

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Lindenstrauss, J., Rosenthal, H.P. The ℒ p spaces. Israel J. Math. 7, 325–349 (1969). https://doi.org/10.1007/BF02788865

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