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The Long Homology Sequence for Quasi-Banach Spaces, with Applications

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Abstract

We establish the existence of long homology sequences in the category of quasi-Banach spaces, with values in a certain category of topological vector spaces. We derive from that new results about the structure of twisted sums of quasi-Banach and Banach spaces. Sample result: let A and B be subspaces of Lp, with 0 < p < 1. If Lp/A and Lp/B are isomorphic then so are A* and B*. In particular, if A and B are finite dimensional, then Lp/A and Lp/B are isomorphic if and only if dim(A) = dim(B).

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References

  1. Aleksandrov A.B.: Essays on non Locally Convex Hardy Classes, in: Havin, V.P and Nikol’skii, N.K. (eds.), Complex Analysis and Spectral Theory, Lecture Notes in Mathematics 864, Springer-Verlag 1981, pp. 1.89

  2. F. Cabello Sánchez J.M.F. Castillo (2000) ArticleTitleDuality and twisted sums of Banach spaces J. Funct. Anal. 175 1–16

    Google Scholar 

  3. Castillo J.M.F., González M. (1997). Three-space problems in Banach space theory, Lecture Notes in Mathematics 1667, Springer-Verlag

  4. S. Dierolf (1973) Uber Vererbbarkeitseigenschaften in topologischen Vektorraumen Dissertation Ludwig Maximilians Universitat Munich

    Google Scholar 

  5. P. Enflo J. Lindenstrauss G. Pisier (1975) ArticleTitleOn the “three-space” problem for Hilbert space Mathematica Scand. 36 199–210

    Google Scholar 

  6. Hilton E., Stammbach K. (1970). A course in homological algebra. Graduate Texts in Mathematics 4, Springer-Verlag

  7. W.B. Johnson (1997) ArticleTitleExtensions of c0 Positivity 1 55–74

    Google Scholar 

  8. N. Kalton (1978) ArticleTitleThe three-space problem for locally bounded F-spaces Compositio Mathematica. 37 243–276

    Google Scholar 

  9. N. Kalton (1984) ArticleTitleLocally Complemented subspaces and L p -spaces for 0<p<1 Mathematische Nachrichten 115 71–97

    Google Scholar 

  10. N. Kalton N.T. Peck (1979) ArticleTitleTwisted sums of sequence spaces and thee three-space problem Trans. Am. Math. Soc. 255 1–30

    Google Scholar 

  11. N. Kalton N.T. Peck (1979) ArticleTitleQuotients of L p (0, 1) for 0 ≤ p < 1 Studia Mathematica. 64 65–75

    Google Scholar 

  12. Kalton N., Peck N.T., Roberts W. An F-space sampler, London Mathematical Society Lecture Note Series 89

  13. N. Kalton W. Roberts (1983) ArticleTitleUniformly exhaustive submeasures and nearly additive set Trans. Am. Math. Soc. 278 803–816

    Google Scholar 

  14. N. Kalton A. Pelczyński (1997) ArticleTitleKernels of surjections from L1-spaces with an application to Sidon sets Mathematische Annalen 309 135–158

    Google Scholar 

  15. Mac Lane S. (1975). Homology, Grundlehren der mathematischen Wissenschaften 114, Springer- Verlag

  16. M. Ribe (1979) ArticleTitleExamples for the nonlocally convex three-space problem Proc. Am. Math Soc. 73 351–355

    Google Scholar 

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Correspondence to Félix Cabello Sánchez.

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Sánchez, F.C., Castillo, J.M.F. The Long Homology Sequence for Quasi-Banach Spaces, with Applications. Positivity 8, 379–394 (2004). https://doi.org/10.1007/s11117-002-2465-y

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