A note on nice operators

In this note, we characterize nice operators in a class of Banach spaces, which includes spaces and L₁(μ), as those operators that preserve extreme points.     

Semisummands and semiideals in banach spaces

Let |·| be a fixed absolute norm onR ². We introduce semi-|·|-summands (resp. |·|-summands) as a natural extension of semi-L-summands (resp.L-summands). We prove that the following statements are equivalent. (i) Every semi-|·|-summand is a |·|-summand, (ii) (1, 0) is not a vertex of the closed unit ball ofR ² with the norm |·|. In particular semi-L ^p-summands areL ^p-summands whenever 1<p≦∞. The concept of semi-|·|-ideal (resp. |·|-ideal) is introduced in order to extend the one of semi-M-ideal (resp.M-ideal). The following statements are shown to be equivalent. (i) Every semi-|·|-ideal is a |·|-ideal, (ii) every |·|-ideal is a |·|-summand, (iii) (0, 1) is an extreme point of the closed unit ball ofR ² with the norm |·|. From semi-|·|-ideals we define semi-|·|-idealoids in the same way as semi-|·|-ideals arise from semi-|·|-summands. Proper semi-|·|-idealoids are those which are neither semi-|·|-summands nor semi-|·|-ideals. We prove that there is a proper semi-|·|-idealoid if and only if (1, 0) is a vertex and (0, 1) is not an extreme point of the closed unit ball ofR ² with the norm |·|. So there are no proper semi-L ^p-idealoids. The paper concludes by showing thatw*-closed semi-|·|-idealoids in a dual Banach space are semi-|·|-summands, so no new concept appears by predualization of semi-|·|-idealoids.     

A note on extreme points in dual spaces

Given a normed space X it can be easily proven that every extreme point in B _X *, the unit ball of X*, is the restriction of an extreme point in B _X ***. Our purpose is to study when the restrictions of extreme points in B _X *** are extreme points in B _X *. Namely, we characterize L ₁-preduals satisfying the aforementioned property.     

The Convex Hull of Extremal Vector-Valued Continuous Functions

Let T be a completely regular space and X a strictly convexn-dimensional real space. We prove that every continuous functionfrom T into the closed unit ball of X can be expressed as anaverage of eight continuous functions from T into the sphereof X if and only if dim (T) n–1, where dim(T) denotesthe covering dimension of T. The proof we give can be used toprove the same fact, without hypotheses on T, when X is infinite-dimensional,although in this case it has been proved recently that a betterresult can be obtained.