Two remarks on diameter 2 properties/Kaks markust diameeter-2 omaduste kohta

A Banach space is said to have the diameter 2 property if the diameter of every nonempty relatively weakly open subset of its unit ball equals 2. In a paper by Abrahamsen, Lima, and Nygaard (Remarks on diameter 2 properties. J. Conv. Anal., 2013, 20, 439-452), the strong diameter 2 property is introduced and studied. This is the property that the diameter of every convex combination of slices of its unit ball equals 2. It is known that the diameter 2 property is stable by taking [l.sub.p]-sums for 1 [less than or equal to] p [less than or equal to] [infinity]. We show the absence of the strong diameter 2 property on [l.sub.p]-sums of Banach spaces when 1 < p < [infinity]. This confirms the conjecture of Abrahamsen, Lima, and Nygaard that the diameter 2 property and the strong diameter 2 property are different. We also show that the strong diameter 2 property carries over to the whole space from a non-zero M-ideal. Key words: diameter 2 property, slice, relatively weakly open set. On toestatud, et artiklis [1] vaadeldud diameeter-2 omadus ja tugev diameeter-2 omadus on erinevad. On naidatud, kuidas diameeter-2 omadused kanduvad M-ideaalilt kogu ruumile.