2018/06/27 by Trond A. Abrahamsen, Abrahamsen, Trond Arnold, Julio Becerra Guerrero +7
Mathematics · #46B04 #46B20 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.1806.10693
openalex publication_date 2018/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and study Banach spaces which have property CWO, i.e., every\nfinite convex combination of relatively weakly open subsets of their unit ball\nis open in the relative weak topology of the unit ball. Stability results of\nsuch spaces are established, and we introduce and discuss a geometric\ncondition---property (co)---on a Banach space. Property (co) essentially says\nthat the operation of taking convex combinations of elements of the unit ball\nis, in a sense, an open map. We show that if a finite dimensional Banach space\nX has property (co), then for any scattered locally compact Hausdorff space\nK, the space C0(K,X) of continuous X-valued functions vanishing at\ninfinity has property CWO. Several Banach spaces are proved to possess this\ngeometric property; among others: 2-dimensional real spaces, finite dimensional\nstrictly convex spaces, finite dimensional polyhedral spaces, and the complex\nspace \ℓ1n. In contrast to this, we provide an example of a\n3-dimensional real Banach space X for which C0(K,X) fails to have\nproperty CWO. We also show that c0-sums of finite dimensional Banach spaces\nwith property (co) have property CWO. In particular, this provides examples of\nsuch spaces outside the class of C0(K,X)-spaces.\n