Haar negligibility of positive cones in Banach spaces

Esterle, J.; Matheron, É.; Moreau, P.

doi:10.1090/spmj/1414

Haar negligibility of positive cones in Banach spaces
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by J. Esterle, É. Matheron and P. Moreau

St. Petersburg Math. J. 27 (2016), 731-756

DOI: https://doi.org/10.1090/spmj/1414

Published electronically: July 26, 2016

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Abstract:

The Haar negligibility of the positive cone associated with a basic sequence is discussed in the case of a separable Banach space. In particular, it is shown that, up to equivalence, the canonical basis of $c_0$ is the only normalized subsymmetric unconditional basic sequence whose positive cone is not Haar null, and the only normalized unconditional basic sequence whose positive cone contains a translate of every compact set. It is also proved that an unconditional basic sequence with a non-Haar null positive cone must be $c_0$-saturated in a very strong sense, and that every quotient of the space generated by such a sequence is $c_0$-saturated.

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