2016/07/13 by Daniel Freeman, Edward Odell, Freeman, D. +5
Mathematics · #Advanced Banach Space Theory #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1607.03587
openalex publication_date 2016/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibit a striking resemblance to the geometry of James' space. Further, we show that the averaging projections onto subspaces spanned by constant coefficient blocks with no gaps between supports are bounded. As a consequence, every Banach space with a spreading basis contains a complemented subspace with an unconditional basis. This gives an affirmative answer to a question of H. Rosenthal. The second part contains two results on Banach spaces X whose asymptotic structures are closely related to c0 and do not contain a copy of ℓ1: i) Suppose X has a normalized weakly null basis (xi) and every spreading model (ei) of a normalized weakly null block basis satisfies ‖e1-e2‖=1. Then some subsequence of (xi) is equivalent to the unit vector basis of c0. This generalizes a similar theorem of Odell and Schlumprecht, and yields a new proof of the Elton-Odell theorem on the existence of infinite (1+ε)-separated sequences in the unit sphere of an arbitrary infinite dimensional Banach space. ii) Suppose that all asymptotic models of X generated by weakly null arrays are equivalent to the unit vector basis of c0. Then X^* is separable and X is asymptotic-c0 with respect to a shrinking basis (yi) of Y⊇ X.