On ideals and generalized centers of finite sets in Banach spaces

In this short note we are interested in studying the relation between the notion of generalized centers of finite sets and the notion of an ideal developed by Godefroy et al. (1993) in [4]. Motivated by some recent work of Veselý (2012) [10], we show that for a Banach space X such that X ∗ is isometric to L 1 ( μ ) for a positive measure μ , if Y ⊂ X is a closed subspace such that for every x ∉ Y , Y ⊂ span { x , Y } is an ideal, then Y has generalized centers for finite sets and is also an ideal in X . For the case of ℓ 1 , we show that if Y ⊂ ℓ 1 satisfies the finite intersection property, and is an ideal in span { x , Y } for all x ∉ Y , then Y is the range of a projection of norm 1 in ℓ 1 . This is related to a well known problem of Lindenstrauss on subspaces which are ranges of projections of norm 1.