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Nests of subspaces in Banach space and their order types

Authors: Alvaro Arias and Jeff Farmer
Journal: Trans. Amer. Math. Soc. 331 (1992), 113-130
MSC: Primary 46B20
MathSciNet review: 1050084
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Abstract: This paper addresses some questions which arise naturally in the theory of nests of subspaces in Banach space. The order topology on the index set of a nest is discussed, as well as the method of spatial indexing by a vector; sufficient geometric conditions for the existence of such a vector are found. It is then shown that a continuous nest exists in any Banach space.

Applications and examples follow; in particular, an extension of the Volterra nest in $ {L^\infty }[ {0,1} ]$ to a continuous one, a continuous nest in a Banach space having no two elements isomorphic to one another, and a characterization of separable $ {\mathcal{L}_p}$-spaces in terms of nests.

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  • [ALWW] G. D. Allen, D. R. Larson, J. D. Ward, and G. Woodward, Similarity of nests in $ {L_1}$, J. Funct. Anal. 92 (1990), 49-76. MR 1064686 (91h:47046)
  • [JRZ] W. B. Johnson, H. P. Rosenthal and M. Zippin, On bases, finite-dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506. MR 0280983 (43:6702)
  • [KS] R. V. Kadison and I. M. Singer, Triangular operator algebras, Amer. J. Math. 82 (1960). MR 0121675 (22:12409)
  • [L1] D. Larson, On similarity of nests in Hilbert space and Banach spaces, Longhorn lecture notes, Springer-Verlag, New York (to appear). MR 967097 (90a:47110)
  • [L2] -, Triangularity in operator algebras, Surveys of Recent Results in Operator Theory (J. Conway, Ed.), Pitman Research Notes in Math., Longman (to appear).
  • [LP] J. Lindenstrauss and A. Pelczyński, Absolutely summing operators in $ {\mathcal{L}_p}$-spaces and their applications, Studia Math. 29 (1968), 275-326. MR 0231188 (37:6743)
  • [LT] J. Lindenstrauss and L. Tzafrifri, Classical Banach spaces. II: Function spaces, Springer-Verlag, Berlin, 1979. MR 540367 (81c:46001)
  • [NW] N. Nielsen and P. Wojtaszczyk, A remark on bases in $ {\mathcal{L}_p}$-spaces with an application of complementary universal $ {\mathcal{L}_\infty }$-spaces, Bull. Acad. Sci. Polon. 21 (1973), 249-254. MR 0322484 (48:846)
  • [R] J. R. Ringrose, Super-diagonal forms for compact linear operators, Proc. London Math. Soc. (3) 12 (1962), 367-374. MR 0136998 (25:458)
  • [W] D. J. Westwood, Nests in Banach space, preprint. MR 1103029 (92c:47055)

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Article copyright: © Copyright 1992 American Mathematical Society

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