The Tsirelson Space 𝒯(p) Has a Unique Unconditional Basis up to Permutation for 0<p<1

Abstract

We show that the $p$-convexified Tsirelson space ${𝒯}^{\left(p\right)}$ for $0<p<1$ and all its complemented subspaces with unconditional basis have unique unconditional basis up to permutation. The techniquesinvolved in the proof are different from the methods that have been used in all the other uniqueness results in the nonlocally convex setting.