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

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

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
2. D. Ills, “Fredgolmovy struktury”, UMN, 26:6(162) (1971), 213–240
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
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5. A. Pełczyński, “On some problems of Banach”, Russian Math. Surveys, 28:5 (1973), 67–76
6. Niels Jørgen Nielsen, “On the Orlicz function spacesL M (0, ∞)”, Isr J Math, 20:3-4 (1975), 237
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
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12. E. M. Semenov, A. M. Shteinberg, “Norm estimates of operator blocks in Banach lattices”, Math. USSR-Sb., 54:2 (1986), 317–333
13. Manuel Gonzalez, “On essentially incomparable Banach spaces”, Math Z, 215:1 (1994), 621
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
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18. S. V. Astashkin, F. A. Sukochev, “Independent functions and the geometry of Banach spaces”, Russian Math. Surveys, 65:6 (2010), 1003–1081
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
20. Alexander Brudnyi, “On the completion problem for algebra”, Journal of Functional Analysis, 2014
21. A. Brudnyi, “Oka principle on the maximal ideal space of $H^\infty$”, Algebra i analiz, 31:5 (2019), 24–89
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