Nests of subspaces in Banach space and their order types

Arias, Alvaro; Farmer, Jeff

doi:10.1090/S0002-9947-1992-1050084-5

Nests of subspaces in Banach space and their order types
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by Alvaro Arias and Jeff Farmer

Trans. Amer. Math. Soc. 331 (1992), 113-130

DOI: https://doi.org/10.1090/S0002-9947-1992-1050084-5

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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.

References

G. D. Allen, D. R. Larson, J. D. Ward, and G. Woodward, Similarity of nests in $\textbf {L}_1$, J. Funct. Anal. 92 (1990), no. 1, 49–76. MR 1064686, DOI 10.1016/0022-1236(90)90067-U
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 280983, DOI 10.1007/BF02771464
Richard V. Kadison and I. M. Singer, Triangular operator algebras. Fundamentals and hyperreducible theory, Amer. J. Math. 82 (1960), 227–259. MR 121675, DOI 10.2307/2372733
David R. Larson, On similarity of nests in Hilbert space and in Banach spaces, Functional analysis (Austin, TX, 1986–87) Lecture Notes in Math., vol. 1332, Springer, Berlin, 1988, pp. 179–194. MR 967097, DOI 10.1007/BFb0081620

Triangularity in operator algebras

J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 231188, DOI 10.4064/sm-29-3-275-326
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
N. J. Nielsen and P. Wojtaszczyk, A remark on bases in ${\cal L}_{p}$-spaces with an application to complementably universal ${\cal L}_{\infty }$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 249–254 (English, with Russian summary). MR 322484
J. R. Ringrose, Super-diagonal forms for compact linear operators, Proc. London Math. Soc. (3) 12 (1962), 367–384. MR 136998, DOI 10.1112/plms/s3-12.1.367
Derek J. Westwood, Nests in Banach space, J. Math. Anal. Appl. 156 (1991), no. 2, 558–567. MR 1103029, DOI 10.1016/0022-247X(91)90414-U