Counterexample to a problem of Geoghegan-West
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- by P. V. Semenov PDF
- Proc. Amer. Math. Soc. 124 (1996), 939-943 Request permission
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
- J. West, Open problems in infinite dimensional topology, preprint, 1990.
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- B. S. Mitjagin, The homotopy structure of a linear group of a Banach space, Uspehi Mat. Nauk 25 (1970), no. 5(155), 63–106 (Russian). MR 0341523
- Raymond Y. T. Wong, On homeomorphisms of certain infinite dimensional spaces, Trans. Amer. Math. Soc. 128 (1967), 148–154. MR 214040, DOI 10.1090/S0002-9947-1967-0214040-4
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Additional Information
- P. V. Semenov
- Affiliation: 1614. app149, Zelenograd, Moscow, 103617, Russia
- Email: SEMENOV.MATAN@MPGU.MSK.SU
- Received by editor(s): July 25, 1991
- Communicated by: James E. West
- © Copyright 1996 American Mathematical Society
- 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