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Extensions and liftings of positive linear mappings on Banach lattices


Author: Heinrich P. Lotz
Journal: Trans. Amer. Math. Soc. 211 (1975), 85-100
MSC: Primary 47B55; Secondary 46M10
DOI: https://doi.org/10.1090/S0002-9947-1975-0383141-7
MathSciNet review: 0383141
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Abstract: Let F be a closed sublattice of a Banach lattice G. We show that any positive linear mapping from F into $ {L^1}(\mu )$ or $ C(X)$ for a Stonian space X has a positive norm preserving extension to G. A dual result for positive norm preserving liftings is also established. These results are applied to obtain extension and lifting theorems for order summable and majorizing linear mappings. We also obtain some partial results concerning positive extensions and liftings of compact linear mappings.


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  • [1] H. F. Bohnenblust, A characterization of complex Hilbert spaces, Portugal. Math. 3 (1942), 103-109. MR 4, 247. MR 0008117 (4:247d)
  • [2] D. W. Dean, Direct factors of (AL)-spaces, Bull. Amer. Math. Soc. 71 (1965), 369-371. MR 31 #5057. MR 0180827 (31:5057)
  • [3] A. M. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958), 482-489. MR 22 #12509. MR 0121775 (22:12509)
  • [4] D. B. Goodner, Projections in normed linear spaces, Trans. Amer. Math. Soc. 69 (1950), 89-108. MR 12, 266. MR 0037465 (12:266c)
  • [5] A. Grothendieck, Une caractérisation vectorielle-métrique des espaces $ {L^1}$, Canad. J. Math. 7 (1955), 552-561. MR 17, 877. MR 0076301 (17:877d)
  • [6] H. Jacobs, Ordered topological tensor products, Dissertation, University of Illinois, 1969.
  • [7] S. Kakutani, Some characterizations of Euclidean space, Japan. J. Math. 16 (1939), 93-97. MR 1, 146. MR 0000895 (1:146d)
  • [8] G. Köthe, Hebbare lokalkonvexe Räume, Math. Ann. 165 (1966), 188-195. MR 33 #4651. MR 0196464 (33:4651)
  • [9] J. L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323-326. MR 13, 659. MR 0045940 (13:659e)
  • [10] L. Nachbin, A theorem of Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28-46. MR 11, 369. MR 0032932 (11:369a)
  • [11] A. Pelczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228. MR 23 #A3441. MR 0126145 (23:A3441)
  • [12] A. L. Peressini, Ordered topological vector spaces, Harper & Row, New York, 1967. MR 37 #3315. MR 0227731 (37:3315)
  • [13] H. H. Schaefer, Topological vector spaces, Macmillan, New York, 1966. MR 33 #1689. MR 0193469 (33:1689)
  • [14] U. Schlotterbeck, Über Klassen majorisierbarer Operatoren auf Banachverbänden, Rev. Acad. Ci. Zaragoza 26 (1971), 585-614. MR 0320809 (47:9343)
  • [15] Z. Semadeni, Projectivity, injectivity and duality, Rozprawy Mat. 35 (1963), 47 pp. MR 27 #4776. MR 0154832 (27:4776)
  • [16] -, Banach spaces of continuous functions. Vol. I, Monografie Mat., Tom 55, PWN, Warsaw, 1971. MR 45 #5730.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0383141-7
Keywords: Extensions and liftings of positive linear mappings, injective Banach lattices, order summable mappings, majorizing mappings, tensor products of Banach lattices
Article copyright: © Copyright 1975 American Mathematical Society

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