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Uspekhi Mat. Nauk, 1970, Volume 25, Issue 5(155), Pages 63–106 (Mi umn5403)  

This article is cited in 21 scientific papers (total in 21 papers)

The homotopy structure of the linear group of a Banach space

B. S. Mityagin


Abstract: The homotopy type of the linear group of an infinite-dimensional Banach space is as important in the theory of Banach manifolds and bundles as is the (stable) homotopy structure of the orthogonal and unitary groups in the theory of finite-dimensional vector bundles and in $K$-theory (for more details see [4]). Kuiper has proved [20] the contractibility of the linear group $GL(H)$ of a Hilbert space $H$, and Neubauer has given a positive answer [34] to the question of the contractibility of $GL(l^p)$, $1\leq p<\infty$, and $gl(c_0)$. At the same time there are examples (the first of which was given by Douady [11]) of Banach spaces with a non-contractible and disconnected linear group. In [30] the author drew attention to the fact that the constructions of Kuiper and Neubauer could be formalized to provide a general procedure for proving (or analysing) the contractibility of the linear group $GL(X)$. This enables us to settle the question of the homotopy structure of the linear groups of many specific Banach spaces. The present paper reviews the results that have been obtained up till now on the contractibility of the linear group of Banach spaces. In § 1 examples are given of Banach spaces with homotopically non-trivial linear groups. The general procedure for analysing the contractibility of $GL(X)$, Theorem 1, is set out in § 2, and the problem of obtaining explicit analytic conditions necessary for the applicability of this procedure is solved in § 3. In §§ 4–6 examples are given of many specific Banach spaces (of smooth and of measurable functions), and the contractibility of their linear groups is proved. § 7 contains remarks on the general procedure and unsolved questions.

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English version:
Russian Mathematical Surveys, 1970, 25:5, 59–103

Bibliographic databases:

UDC: 519.4+519.5
MSC: 55Q45, 55Q40, 47L10, 55R25, 55R50, 46E30
Received: 18.12.1969

Citation: B. S. Mityagin, “The homotopy structure of the linear group of a Banach space”, Uspekhi Mat. Nauk, 25:5(155) (1970), 63–106; Russian Math. Surveys, 25:5 (1970), 59–103

Citation in format AMSBIB
\Bibitem{Mit70}
\by B.~S.~Mityagin
\paper The homotopy structure of the linear group of a~Banach space
\jour Uspekhi Mat. Nauk
\yr 1970
\vol 25
\issue 5(155)
\pages 63--106
\mathnet{http://mi.mathnet.ru/umn5403}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=341523}
\zmath{https://zbmath.org/?q=an:0232.47046}
\transl
\jour Russian Math. Surveys
\yr 1970
\vol 25
\issue 5
\pages 59--103
\crossref{https://doi.org/10.1070/RM1970v025n05ABEH003814}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. B. S. Mityagin, I. S. Edel'shtein, “Homotopy type of linear groups of two classes of Banach spaces”, Funct. Anal. Appl., 4:3 (1970), 221–231  mathnet  crossref  mathscinet  zmath
    2. D. Ills, “Fredgolmovy struktury”, UMN, 26:6(162) (1971), 213–240  mathnet  mathscinet  zmath
    3. V. I. Arkin, V. L. Levin, “Convexity of values of vector integrals, theorems on measurable choice and variational problems”, Russian Math. Surveys, 27:3 (1972), 21–85  mathnet  crossref  mathscinet  zmath
    4. B. S. Mityagin, “On idempotent multipliers in symmetric functional spaces”, Funct. Anal. Appl., 6:3 (1972), 244–245  mathnet  crossref  mathscinet  zmath
    5. A. Pełczyński, “On some problems of Banach”, Russian Math. Surveys, 28:5 (1973), 67–76  mathnet  crossref  mathscinet  zmath
    6. Niels Jørgen Nielsen, “On the Orlicz function spacesL M (0, ∞)”, Isr J Math, 20:3-4 (1975), 237  crossref  mathscinet  zmath
    7. M. G. Zaidenberg, S. G. Krein, P. A. Kuchment, A. A. Pankov, “Banach bundles and linear operators”, Russian Math. Surveys, 30:5 (1975), 115–175  mathnet  crossref  mathscinet  zmath
    8. D. Alspach, Y. Benyamini, “Primariness of spaces of continuous functions on ordinals”, Isr J Math, 27:1 (1977), 64  crossref  mathscinet  zmath
    9. Yu. G. Borisovich, V. G. Zvyagin, Yu. I. Sapronov, “Non-linear Fredholm maps and the Leray–Schauder theory”, Russian Math. Surveys, 32:4 (1977), 1–54  mathnet  crossref  mathscinet  zmath
    10. A. V. Bukhvalov, A. I. Veksler, G. Ya. Lozanovskii, “Banach lattices – some Banach aspects of their theory”, Russian Math. Surveys, 34:2 (1979), 159–212  mathnet  crossref  mathscinet  zmath
    11. Israel Aharoni, Joram Lindenstrauss, “An extension of a result of ribe”, Isr J Math, 52:1-2 (1985), 59  crossref  mathscinet  zmath  isi
    12. E. M. Semenov, A. M. Shteinberg, “Norm estimates of operator blocks in Banach lattices”, Math. USSR-Sb., 54:2 (1986), 317–333  mathnet  crossref  mathscinet  zmath
    13. Manuel Gonzalez, “On essentially incomparable Banach spaces”, Math Z, 215:1 (1994), 621  crossref  mathscinet  zmath  isi
    14. P. G. Dodds, F. A. Sukochev, “Contractibility of the linear group in Banach spaces of measurable operators”, Integr equ oper theory, 26:3 (1996), 305  crossref  mathscinet  zmath  isi
    15. F. A. Sukochev, “Linear-topological classification of separableL p-spaces associated with von Neumann algebras of type I”, Isr J Math, 115:1 (2000), 137  crossref  mathscinet  zmath  isi
    16. S. V. Astashkin, F. A. Sukochev, “Sums of independent functions in symmetric spaces with the Kruglov property”, Math. Notes, 80:4 (2006), 593–598  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    17. S. V. Astashkin, F. A. Sukochev, “Series of independent mean zero random variables in rearrangement invariant spaces with the Kruglov property”, J. Math. Sci. (N. Y.), 148:6 (2008), 795–809  mathnet  crossref  mathscinet  elib  elib
    18. S. V. Astashkin, F. A. Sukochev, “Independent functions and the geometry of Banach spaces”, Russian Math. Surveys, 65:6 (2010), 1003–1081  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    19. Alexander Brudnyi, “Holomorphic Banach vector bundles on the maximal ideal space of and the operator corona problem of Sz.-Nagy”, Advances in Mathematics, 232:1 (2013), 121  crossref
    20. Alexander Brudnyi, “On the completion problem for algebra”, Journal of Functional Analysis, 2014  crossref
    21. A. Brudnyi, “Oka principle on the maximal ideal space of $H^\infty$”, Algebra i analiz, 31:5 (2019), 24–89  mathnet
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