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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Non-ergodic Banach spaces are near Hilbert


Author: W. Cuellar Carrera
Journal: Trans. Amer. Math. Soc. 370 (2018), 8691-8707
MSC (2010): Primary 46B20, 46B03; Secondary 03E15
DOI: https://doi.org/10.1090/tran/7319
Published electronically: September 10, 2018
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Abstract: We prove that a non-ergodic Banach space must be near Hilbert. In particular, $ \ell _p$ ( $ 2<p<\infty $) is ergodic. This reinforces the conjecture that $ \ell _2$ is the only non-ergodic Banach space. As an application of our criterion for ergodicity, we prove that there is no separable Banach space which is complementably universal for the class of all subspaces of $ \ell _p$, for $ 1\leq p <2$. This solves a question left open by W. B. Johnson and A. Szankowski in 1976.


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Additional Information

W. Cuellar Carrera
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, R. do Matão 1010 SP-Brazil
Email: cuellar@ime.usp.br

DOI: https://doi.org/10.1090/tran/7319
Received by editor(s): December 27, 2016
Received by editor(s) in revised form: June 17, 2017, and June 26, 2017
Published electronically: September 10, 2018
Additional Notes: The author was supported by FAPESP grant 2014/25900-7.
Article copyright: © Copyright 2018 American Mathematical Society

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