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Counterexample to a problem of Geoghegan-West


Author: P. V. Semenov
Journal: Proc. Amer. Math. Soc. 124 (1996), 939-943
MSC (1991): Primary 58B05, 57N20
DOI: https://doi.org/10.1090/S0002-9939-96-03276-5
MathSciNet review: 1317050
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Abstract: Let $X$ be a Banach space and $\mathrm{GL}(X)$ its general linear group. Let $\|\cdot\|$ denote the operator norm and ``$w$'' the pointwise convergence topology on $\mathrm{GL}(X)$. Is the identity map $(\mathrm{GL}(X),\|\cdot\|)\rightarrow(\mathrm{GL}(X),w)$ a homotopy equivalence? The answer is negative. One of the possible counterexamples is a well-known James space $\mathbb J$---the ``space of counterexamples in Banach spaces theory''.


References [Enhancements On Off] (What's this?)

  • 1. J. West, Open problems in infinite dimensional topology, preprint, 1990.
  • 2. R. Geoghegan, Open problems in infinite dimensional topology, Topology Proc. 4 (1979), 287--330. MR 82a:57015
  • 3. B. S. Mitjagin, Homotopic structure of linear groups of Banach spaces, Uspekhi Mat. Nauk 25 (1970), no. 5, 63--106. MR 49:6274a
  • 4. R. Wong, On homeomorphisms of certain infinite dimensional spaces, Trans. Amer. Math. Soc. 128 (1967), 148--154. MR 35:4892
  • 5. R. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518--527. MR 12:6166

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Additional Information

P. V. Semenov
Affiliation: 1614. app149, Zelenograd, Moscow, 103617, Russia
Email: SEMENOV.MATAN@MPGU.MSK.SU

DOI: https://doi.org/10.1090/S0002-9939-96-03276-5
Keywords: General linear group, pointwise convergence topology, weak homotopy equivalence, James space
Received by editor(s): July 25, 1991
Communicated by: James E. West
Article copyright: © Copyright 1996 American Mathematical Society

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