Uniqueness of unconditional and symmetric structures in finite dimensional spaces (Q919569)

This paper is a study of uniqueness of symmetric and unconditional bases in a family of finite dimensional Banach spaces (see the paper for a precise definition of these concepts). In general, the results obtained are too highly technical in nature and content to be detailed here. However a major application worth mentioning is that of a rearrangement invariant Banach function space X on [0,1]. In this case, letting \(X_ n\) be those functions which are constant on each atom of the algebra \(B_ n\) generated by the interval [(k-1)/n,k/n), \(1\leq k\leq n\), the family of subspaces \(\{X_ n\}^{\infty}_{n=1}\) is shown to have a unique symmetric basis and an almost unique unconditional basis.