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I have two quick questions:

It can be shown without too much trouble (using methods from Altshuler/Casazza/Lin, 1973) that any Lorentz sequence space admits a unique (up to equivalence) subsymmetric basis (which is also symmetric).

Question 1. Are there any other known examples of a Banach space admitting a unique subsymmetric basis?

In 2004, Sari showed that the Tirilman spaces admit a subsymmetric basis but not a symmetric one.

Question 2. Are there any other known examples of a Banach space admitting a subsymmetric basis but not a symmetric basis?

I am especially interested in Banach spaces satisfying both properties simultaneously---that is, a Banach space admitting a unique subsymmetric basis which is not symmetric.

Thanks!

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  • $\begingroup$ Did you check the literature on Schlumprecht's space? $\endgroup$
    – Bill Johnson
    Commented Oct 11, 2016 at 22:13
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    $\begingroup$ @BillJohnson Yes I did, and I almost mentioned it above. It was asked in the "Remarks" paper (Kutzarova/Lin, 1998) whether Schlumprecht space $S$ has a unique subsymmetric basis, but to my knowledge that question has never been answered. Nor do I know whether it has ever been investigated to see if $S$ has a symmetric basis (which would give a negative answer to the uniqueness question). $\endgroup$
    – Ben W
    Commented Oct 11, 2016 at 22:25

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