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Conditional expectations and an isomorphic characterization of $ L\sb{1}$-spaces


Author: L. Tzafriri
Journal: Proc. Amer. Math. Soc. 27 (1971), 317-324
MSC: Primary 46.35; Secondary 28.00
DOI: https://doi.org/10.1090/S0002-9939-1971-0275143-8
MathSciNet review: 0275143
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Abstract: Conditional expectations can be defined in Banach spaces whose elements can be represented as measurable functions. In the present paper it is shown that such a space (precisely a cyclic space) is isomorphic to an $ {L_1}$-space if and only if the conditional expectations act as bounded operators for sufficiently many representations.


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  • [1] T. Ando, Banachverbände und positive Projektionen, Math. Z. 109 (1969), 121-130. MR 0511673 (58:23480)
  • [2] W. G. Bade, On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 (1955), 345-360. MR 17, 513. MR 0073954 (17:513d)
  • [3] -, A multiplicity theory for Boolean algebras of projections in Banach spaces, Trans. Amer. Math. Soc. 92 (1959), 508-530. MR 21 #7443. MR 0108729 (21:7443)
  • [4] N. Dunford, Spectral operators, Pacific J. Math. 4 (1954), 321-354. MR 16, 142. MR 0063563 (16:142d)
  • [5] H. W. Ellis and I. Halperin, Function spaces determined by a levelling length function, Canad. J. Math. 5 (1953), 576-592. MR 15, 439. MR 0058869 (15:439c)
  • [6] N. E. Gretsky, Representation theorems on Banach function spaces, Bull. Amer. Math. Soc. 74 (1968), 705-709. MR 37 #2009. MR 0226419 (37:2009)
  • [7] M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $ {L_p}$, Studia Math. 21 (1961/62), 161-176. MR 27 #2851. MR 0152879 (27:2851)
  • [8] W. A. J. Luxemburg, Rearrangement-invariant Banach function spaces, Proc. Sympos. Anal. (Queen's Univ., 1967), Queen's Papers in Pure and Appl. Math., no. 10, 1967, pp. 88-144.
  • [9] W. A. J. Luxemburg and A. C. Zaanen, Notes on Banach function spaces. I, Nederl. Akad. Wetensch. Proc. Ser. A 66=Indag. Math. 25 (1963), 135-147. MR 26 #6723a. MR 0149231 (26:6723a)
  • [10] I. Singer, Some characterizations of symmetric bases in Banach spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 185-192. MR 26 #5393. MR 0147880 (26:5393)
  • [11] L. Tzafriri, An isomorphic characterization of $ {L_p}$ and $ {c_0}$-spaces, Studia Math. 32 (1969), 287-296.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0275143-8
Keywords: Conditional expectation, Banach function spaces, cyclic spaces, $ {L_1}$-spaces, Boolean algebras of projections, isomorphism between Banach spaces
Article copyright: © Copyright 1971 American Mathematical Society

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