Remarks on diameter 2 properties

2013/04/26 by Trond A. Abrahamsen, Abrahamsen, Trond, Vegard Lima +3 · 6 citations
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1304.7068

Abstract

If X is an infinite-dimensional uniform algebra, if X has the Daugavet property or if X is a proper M-embedded space, every relatively weakly open subset of the unit ball of the Banach space X is known to have diameter 2, i.e., X has the diameter 2 property. We prove that in these three cases even every finite convex combination of relatively weakly open subsets of the unit ball have diameter 2. Further, we identify new examples of spaces with the diameter 2 property outside the formerly known cases; in particular we observe that forming ℓ_p-sums of diameter 2 spaces does not ruin diameter 2 structure.

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