Journal of Convex Analysis 22 (2015), No. 1, 001--017
Copyright Heldermann Verlag 2015
Stability Results of Diameter Two Properties
María D. Acosta
Dep. de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
dacosta@ugr.es
Julio Becerra-Guerrero
Dep. de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
juliobg@ugr.es
Ginés López-Pérez
Dep. de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
glopezp@ugr.es
A Banach space has the diameter two property if every (nonempty) weakly open set of its unit ball has diameter two. We prove that this property is stable under finite sums, whenever an absolute norm is considered in the product space, improving some previous results. Recently T. A. Abrahamsen, V. Lima and O. Nygaard [J. Convex Analysis 20 (2013) 329--338] defined the so-called strong diameter two property, i.e. every convex combination of slices in the unit ball has diameter two.
We show that the strong diameter two property is never stable under (non-trivial) lp-sums. As a consequence, both diameter two properties are not equivalent. We also prove that any Banach space whose centralizer is infinite-dimensional has the strong diameter two property, solving an open problem. This result can be applied to infinite-dimensional C*-algebras and L1-preduals, for instance.
Keywords: Banach space, weakly open set, slice, absolute norm, diameter two property, Radon-Nikodym property.
MSC: 46B20; 46B22
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